Punta Arenas is almost at the "end of the world" so to speak, so while you will find awesome spots to eat, bear in mind the population in Punta Arenas does not sustain expensive eateries by itself, so everything you find will have some tweaks to meet tourist tastes.
Parilla Los Ganaderos offers a great parilla style restaurant (grilled game, meats, etc.) A lot of people have never eaten lamb before and they mistake it for something similar to beef. It is not, while it is a very tasty type of meat you have to eat it shortly after coming out of the grill. Otherwise the fatty tissue will begin to harden and the meat will began to whisper some sort of spoiled milk taste (I'm exaggerating, but somewhat like that). Also, this meat once cooked will NEVER taste the same if you reheat it. So if you are going to order lamb, make sure it has just left the grill or that it is specially cooked for you. Otherwise you will not have the same experience and therefore you'll read some of the critiques below.
If you don't want lamb, no one eats lamb twice in a week. They have beef, pork, fish and chicken dishes. One thing not to miss (here or anywhere in Chile) is to open your order with empanadas, either cheese or ground beef, etc. You'll love them.
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