Hobbit food is NOT American, Mexican, Mediterranean, Middle Eastern etc.
New World Foodstuffs:
Tolkien specifically ruled out any New World foods, with the big exception of potatoes, (and pipeweed/tobacco) for Hobbits. Potatoes are allowed, Tolkien decided that they were such a staple of the English diet they had to be included, so they are referred to as taters and it opens the possibility that canonically they are NOT potatoes, but an identical kind of tuber. For our purposes, potatoes are in.
Tomatoes are out. Sorry. Yes, part of the quintessential English breakfast, amongst other things, but Tolkien ditched them.
All other New World foodstuffs are to be avoided, so no:
Maize (for you Americans out there, there are repeated references to corn in the Books, corn is a generic term for grains, generally it means wheat)
None of it, sorry. Tropical fruits did not become common in England until after WWII. Even today pineapple is considered very exotic.
No garlic, no European mustards (English mustard is fine) and cheeses, nuts, salamis, etc.
No olive oil. In fact, no oil, Hobbits cook with butter.
No foccacia and bruschetta etc.
The above all apply to all Middle Earth too, not just the Shire. There is no New World or tropics in Middle Earth.
Problematic. The English did of course eat seafood, minus such things as prawns and calamari. But not so much inland, and the Shire is unfortunately landlocked and some distance from the sea. I personally would accept sea fish as an occasional Hobbit foodsource, perhaps imported and generally salted or smoked, not fresh. Oysters, mussels, winkles etc. were a part of traditional English cuisine and may be included in moderation, though I do not believe there is any canonical support for them. Discretion required.
There is a quote from Tolkien during an interview where he was asked about what Hobbits would eat, and he said something along the lines of "none of that cuisine mystique." So take him at his word. No aioli and emulsions.