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His cooking is the real deal — trained in the French style.
Literal
His cooking [topic-は] France trained [possessive-の] authentic [copula-だ].
フランス仕込み ('French-trained, schooled in France') is a compound: フランス + 仕込み ('training, preparation' — from 仕込む, 'to train, to prepare'). 本格派 (ほんかくは, 'the real deal, authentic school') means someone who does things properly, not in a half-hearted way. The sentence is an admiring character description — two compact noun compounds doing all the work.