Agree generally with the comments and I do think Taiwan food is hard to categorize. As noted, there are also major influences from many parts of mainland China given the exodus of the Kuomintang (as noted, pinyin is Guo Min Dang, but KMT is still more common reference I think) to Taiwan and there's a Fujian influence as well. And influences from the Japanese occupation, and the indigenous population.
Snacks/street food are one clearly distinctive type of food from Taiwan. Examples such as stinky tofu or oyster omelets are
perhaps most associated with Taiwan, although I
think they more likely originated on the mainland. Same is true for other foods that many associate with Taiwan. E.g., breakfast items like soybean milk, fried dough, shaobing; noodle soups like beef noodle soup; dumpling items, such as xiaolongbao, potstickers and shenjianbao; shaved ice; etc. I don't know enough about the history of all these items to say if any originated in Taiwan (some/most clearly did not), but I think most people from Taiwan think of these foods as closely linked to Taiwan. Tea (e.g., Oolong or Ti Kuan Yin) are also very popular.
Three cup chicken is a great idea of something to make. Relatively straightforward to make and very distinctively from Taiwan (even though, again, I don't think the dish truly originated there).