For the sake of the argument, the "Western" countries should include all Member States of the EU, and countries conventionally understood to be part of the "West".
Any time after World War II.
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Hungary is one of the most prominent examples of a Western country with a single party holding two-thirds majority of parliamentary seats.
It should be noted that this is not without controversey as Orban's governing coalition has been criticized for eroding Hungary's democratic institutions to the degree that elections have essentially become non-competitive.
Freedom House currently ranks Hungary as only "Partly Free" with a score of 69/100. A very unusual score for a EU member state.
Barbados is a Western country with a single party holding all of the seats in Parliament.
This supermajority was used to amend the Barbados constitution in 2021 for the purpose of transitioning from a constitutional monarchy to a republic. Essentially removing Queen Elizabeth II as head of state and making the incumbent Governor-General (a Barbados citizen) as the President.
Unlike Hungary, Barbados does have a functional democratic system, which makes Labour Party's total dominance all the more unusual in Western world. The situation was so unusual that one of the MPs crossed the floor as independent after the 2018 election to be the sole opposition, citing concerns for democracy.
Freedom House currently ranks Barbados as "Free" with a score of 95/100.
The United Kingdom...well...sort of...
The UK doesn't have a set of law that is called "The Constitution" or "Basic Law", instead there are a collection of statutes, traditions and conventions. Among these traditions is "Parliamentary Supremacy", which says that Parliament may pass or repeal any law, including changing the constitution. This it has done on several occasions since World War II:
The changes were made by majorities in Parliament, and all are clearly constitutional changes.
A constitutional change can be implemented by a simple majority in Parliament, which is the usual situation as a result of the FPTP electoral system. There is another tradition that significant constitutional changes must also be passed by referendum, but it is up to Parliament to decide what is "significant", and that determination is often political, not legal.
In my opinion the most prominent country that had this is
German coalitions are today generally CDU/CSU + FDP and SPD + the Greens. Though the current coalition contains SPD + the Greens + FDP. Prior to the rise of the Greens and the left there were only 3 parties in Parliament normally: CDU/CSU (technically two parties) SPD and FDP, with the FDP being the so called Kingmaker.
From 1965 to 1966 there was a coalition of CDU/CSU and FDP under Ludwig Erhard, which failed in 1966 leading to a coalition of CDU/CSU and SPD under Kurt Georg Kiesinger which held 468/518 or a bit over 90% of the seats of the Bundestag.The supermajority to change the Grundgesetz is 2/3.
There were 12 changes made to the Grundgesetz in that period, the most interesting ones were the establishment of the so called Notstandsgesetze and reformation of the financial system of the states and the federal goverment.
As @Jörg W Mittag pointed out in the comments, one needs both chambers of the german parliament to change the Grundgesetz, the other chamber being the Bundesrat.
Edit: Apparently one can't split the votes for one Bundesland in the Bundesrat. So my previous calculations weren't correct. I found that many votes in the Bundesrat were unanimously. I found only one abstain vote, though there could very well be more.
Interesting sidenote: This coalition was the first time since 1930 that the SPD was part of the Goverment.
In France, a constitutional revision requires either a referendum, or a 3/5 super-majority in both chambers as a whole (ie. gather the National Assembly and the Senate in the same room, usually in Versailles, then 3/5th of all votes are required, without distinction.)
This majority has been achieved many times, for example, during the 12th legislature, between 2007 and 2012.
The constitution of the Portuguese Republic of 1976 establishes in its article 286 that a constitutional revision needs a qualified majority of two thirds of the deputies to the Republic's Assembly.
All constitutional revisions since 1976 have needed the approval of the two largest parties, the Socialist Party (PS) and the Social-Democratic Party (PSD). There were 7 revisions, 1982, 1987, 1992, 1997, 2001, 2004 and 2005, but none of the constitutional revisions happened with both parties sharing the government.
Following the legislative elections of 1983 the PS got 101/250 (40.4%) MP's and the PSD 75/250 (30.0%) MP's. Combined they had 70.4% of the parliamentary seats and formed a government coalition with PS leader Mário Soares as prime-minister and PSD leader Carlos Mota Pinto vice-prime-minister and defense minister.
Though this was the only time a government majority could revise the constitution, it did not do so. A revision had been voted in the previous year and with strong opposition within the two parties, with factions objecting to a coalition with their natural rival, there was no reason why nor opportunity to revise the constitution.
Austria has had many coalitions after World War 2 between the main social-democratic party (SPÖ) and the main Catholic conservative party (ÖVP). Since those are not exactly similar ideologies, these coalitions do not usually get very much done; they usually have lots of public disputes between each other when they are in a coalition.
It used to be the case that relatively few people in Austria voted for anything other than these two, so they had the necessary two-thirds majority to change the federal constitution. However, various other parties have been getting more and more popular over the years, so that is unlikely to happen again any time soon.
From a quick lecture of Wikipedia, this happened most recently between 2006 and 2008, when the SPÖ got 68 seats at the election, the ÖVP 66, summing up to 134, more than the 122 that comprise two thirds.