This might be just me drawing parallels between Italian and other languages, but isn't "la" used when you know exactly which thing you are looking for? For exemple, if I'm looking for a bowl, any bowl, I might say "una ciottola", but if I'm looking for a specific bowl which I was telling my friend about, I would use "la ciottola".
Here, I feel like "Il cuoco ha la crema al cioccolato" would be the answer to a question like "Who has the custard" or "where is the custard" (the one that was prepared, as everyone in the conversation knows which custard they are speaking of: The cook has the custard). The English translation without an article seems like the cook just has some custard nobody knew about...
Not in this case, no. This is just how Italian talks about food. In English, we can use nouns like they're adjectives, so we can say "flavor food" or "ingredient food", but in Italian they can't do that, so they say "food to the flavor" or "food of ingredient".
Even in English, when we use the long-form possessive, we use "the": "start of the day"; "head of the class"; etc.
Oh, it's not the "al cioccolato" that bothers me. It's the sudden disappearance of the article before "crema" in the English translation. English isn't my native language, but I always thought the article could only disappear like that when it doesn't matter which thing it is. (So the answer given in English, "The cook has chocolate cream" would mean he has "some chocolate cream". Meanwhile, the Italian sentence uses a definite article, so I don't understand why the translation isn't "the cook has the chocolate cream"
Anyone can be a cook. It takes special training to be a chef.