Maybe Zero Otto Nove or Eataly if you are willing to brave the lunch crowds.
Blue Smoke has an excellent burger and many other good options on the menu, though you might go over your budget a tiny bit.
116 East 27th Street, New York, NY 10016
200 5th Ave, New York, NY 10010
Zero Otto Nove
15 W 21st St, New York, NY 10010