I spent some time in Taipei around this time last year. I’d second @Night07 's rec for Din Tai Fung. I went to the original branch. This is near Dongman MRT station only 1 stop from Daan MRT. The pork and crab xlb were one of the best things I ate in Taipei. I tried the a Din Tai Fung branch in Kuala Lumpur and it was nowhere near as good. Both in terms of food and service.
I know you don’t really want street food but the beef noodles at Yong Kang were really good. This is also very close to Din Tai Fung.
Like @Night07 I also went to Shin Yeh at Taipei 101. It is a little bit touristy and there’s a minimum charge of 1800 TD but with two people it shouldn’t be a problem ( I had to bump up my bill with wine as I was on my own. The views were great though and considering it’s 500 TD to go up Taipei 101 I decided to get almost as good a view and put the 500TD towards lunch. The food is good but not outstanding but I would go back to have the Steamed Codfish with Taiwanese
Cummingcordia and preserved Mustard Leaf. Perfectly cooked fish and I’d never had cummingcordia before but they were delicious.
I’ll be in Hong Kong in February so will be following this thread with Interest but I think Tasting Court is a definite.