if we are talking about hungarian cuisine don't leave out töltött káposzta (stuffed cabbage) . Also I'm pretty sure hungarian settlers would have brought pálinka (hungarian liquor made from all kinds fruits, most popular i think is plum) with them.
Few things about current hungarian culture but i think many things will change by 2300:
Pessimism - this is probably the most prominent cultural trait in hungary, people usually notice bad things more and we are always suspicious that there is a catch attached to every good new.
Family Centric - Strongest in the countryside. I think extended family is more important in hungary than elsewhere in europe and people don't tend to have huge networks of friends, mostly small friend groups.
Shifty-ness - This one i think is common in other post-communist countries but hungarians are distrustful towards the government and society at large by default. Most hungarians also feel kinda "separated" from their representatives in both local and country level administration and most don't think of the state as the sum of all citizens. This lead to a mentality where people try to get ahead circumventing rules and such, smaller forms of tax evasion are extremely common.
Valuing Independence - I really don't feel like writing this part now sorry
International relations:
Many hungarians have negative thougths about the following:
Romanians, Slovaks and Ukrainians.
Many hungarians have good opinions about:
Polish(though not nearly as much as they like us), Austrians, Croatians(kind of a stretch).
Maybe i'll write more if i feel like it. I hope it helps!