After two consecutive visits to this restaurant (I'll explain later why two) I'm ready to name Feherszarvas Vadasztanya Candidate#1 to my personal Restaurant of the year award (for 2012 it was Kindli in Zurich). So let me detail my impression.
Food. If you want to try Hungarian "wild" food, it's the best place to go. We tried two dishes from venison, rabbit soup, wild piglet soup - and everything was delicious. The menu is big enough to find whatever you like - fish, poultry, meat...
Wine list. You can find a wide selection of Egeri wines; and most of them are very interesting and underestimated. Even more, you can buy the wine you like, and it's worth doing (I personally left the restaurant with 7 bottles). My recommendations are: Bikaver (the most famous Egery wine) - ask for the best year, I'd recommend 2008; and a house wine "Feherszarvas Vadasztanya Szarvasvadasz". Prices of the wine shop are very attractive (as well as of the restaurant itself).
Interior and service. A lot of hunting stuff everywhere, live piano music in the evening, English speaking waiters, menu on a number of foreign languages - and it's still not a "tourist place" - most of visitor are locals.
Location. The restaurant is in 5 min walking distance from Egeri Thermal Baths, so we did a great combination: lunch-bath-dinner :) Even in case you have to drive to different city, it allows you to have a glass of wine during lunch (Hungary has 0 alcohol tolerance while driving).
Overall - nothing to complain, great food, vine and atmospere - a must visit place in Eger.