I was looking at this and it seems European countries are sometimes respondent to WTO disputes, and not the EU as a whole, so I was wondering if the EU has some kind of mechanism to handle these cases before they are brought by the WTO, because I didn't see any European country complain against another European country at the WTO.
EU members have agreed by treaty not to take trade disputes to the WTO. This is one of the effects of the following article of the Union's founding treaties:
(ex Article 292 TEC)
Member States undertake not to submit a dispute concerning the interpretation or application of the Treaties to any method of settlement other than those provided for therein.
EU trade is tightly controlled by the treaties (which is not surprising given the EU's origins), so it is clearly covered by this article.
Disputes between EU member countries are indeed usually handled EU-internally. Usually in the Court of Justice of the European Union. The EU single market is in most areas far more regulated than the international market. So when two EU countries have a trade dispute, that's almost always a dispute about EU-internal rules, and not WTO rules.