So much to see, do and eat in Budapest. For a lunch, don't miss a visit to the Central Market Hall. You may have to be patient to find a seat and it can get crowded, but you'll be rewarded with some delicious food. For me, sausages and potatoes did the trick. You may also want to try the uniquely Budapest fried dough langos. IMO, they're best topped with sour cream and cheese, and drizzled with just a little garlic. If you find the langos line long (make sure to get one freshly made), there's a booth outside near the funicular just across on the Buda side of the river and it's a nice spot to sit down and have a nibble.
I spent some time exploring my Hungarian Jewish heritage. And for dinner, a taste of the old country at Rosenstein Vendeglo. It's a slightly upscale (yet still casual) Jewish restaurant. If I were to recommend two dishes, I'd say the matzoh ball soup and the lamb with potato latke. While the latke itself was nothing special, the lamb was terrific.
Elsewhere, I had a chilled sour cherry soup and crispy roast goose leg with red cabbage and mashed potatoes. Both were excellent . . . and I'm frustrated because while I ordinarily document every place I eat, I can't find where this was. . . . I'll keep searching. Perhaps it was in Prague, but I don't think so.
We had dinner at Costes, a Michelin 1-star restaurant. The food was solid (not great), plated beautifully, and the many local wines elevated the meal. But I had just eaten at La Degustation Bohême Bourgeoise in Prague and enjoyed it quite a bit more. I still lean towards recommending it, but not strongly.
Don't miss the gelato at Gelarto Rosa. And the pastries at Ruszwurm Cukraszda were also excellent. We managed to fit both in between visits to nearby tourist spots so you don't need to veer far off the beaten path. In the Jewish quarter, we made a brief stop for pastries at Frohlich Bakery and Cafe - delicious pastries, including Rugelle, which if I had to guess is a Hungarian Jewish take on what we call rugelach here.
Not everything I ate in Budapest was wonderful. The burger culture in Budapest is strong and we visited the hugely popular Tuning Bar. A cool place and hugely popular, but the burgers were just decent. Also, a small chain hummus bar was certainly convenient but the pita was a tad dry, and the falafel and shakshuka just average. Finally, a slightly upscale outdoor cafe we stumbled upon served a very mediocre version of chicken paprikash. I wanted a more traditional version and ended up with something a bit too fussy.
As for sights, the slightly off the beaten path Memento Park was lightly visited but hugely fascinating. Oversized statues, once adoring this city, now all here serving as a reminder of past days of communism. You can follow the guide books for most everything, but Memento Park might be overlooked. On the other hand, I waited quite a while for the House of Terror and I was not so impressed, but YMMV. I would have preferred to simply read about it and the horrors suffered throughout Hungary.
I wasn't even in Vienna for 24 hours, and most of that time was spent exploring palaces and the like and searching for Mozart . . . such a gorgeous city. But I did not miss the opportunity to visit the extravagant and world famous (and for good reason) Demel. If you plan to eat pastries in Vienna, make sure Demel is on your list. The sacher torte and everything else I tried were stellar.