Séf utcája was one of the first restaurants to take hold in the newly renovated market in Hold utca. I've been there several times trying out different dishes and was never disappointed. They serve mostly traditional Hungarian and Jewish dishes - e.g. pastrami, cholent, lecsó - in high quality. Prices range from 1,500 to 2,100 HUF and portions are quite big. They have a different lunch dish every day, you can find these on their fb page. On weekdays the market closes around 6 I think so you better go for lunch, except for Friday when it's open till late. Local food doesn't get much better than this (even if simple) so it's recommended for tourists too.
This old market has been revived through the development of pop-up restaurants upstairs. They are only open during the day and you do not get five star service, with many of the eateries offering communal bench seating. But there is a good selection of restaurants and the food is both delicious, on the whole authentically Hungarian (there are also Italian and Chinese outlets) and remarkably good value. I had a huge Hungarian sausage, which was served on a large piece of bread with some mustard and a side order of a pickle stuffed with cabbage. The tastes were explosive in my mouth and, together with a large glass of wine, I paid only about £5. The menus change regularly and there are often dishes of the day.
If you want to experience the worst customer service possible, this is a must try! We wanted to order a Vietnamese soup, which is served infront of the restaurant. The woman serving the soup 'explained' (she doesn't speak English) us that we have to pay first inside the restaurant and only then could be served. When you pay you receive tickets, which you have to show at the soup station. We ordered Tom Ka Gai with shrimps. Unfortunately we ordered the wrong soup (there is Pho and Tom ka gai) - as none of the waiters was able to speak English and explain us the difference between the dishes. So we tried to exchange our tickets again to get a Pho soup, which we originally wanted, and which was cheaper than the Tom ka gai. First, the waiters did not want to exchange the tickets - we hadn't even ordered the soup yet - so only the ticket had to be changed. When they finally exchanged the tickets, they didn't want to give us back the extra money we had payed. The women behind the counter were extremely rude - talking in Hungarian about us, rolling their eyes, throwing the tickets on the counter in a very aggressive way. I have never experienced such unpleasant waiters ever before in my life - and I really have difficulties describing this experience - as no words can describe the behavior of the staff at this restaurant. On top, the soup didn't even taste well. You should definitely rather try another restaurant, such as the Italian place next to A Séf Utcája - where the staff is very nice and welcoming and the food is very good as well!