Cut both meats into small pieces. Grind through the large holes of a meat grinder. Combine in a bowl, then grind together through small holes of grinder. Divide into 4 equal parts (each about 5 3/4 ounces). Form into patties, each about 3 1/2 inches in diameter.
Preheat grill to medium-high. Season both sides of burgers with coarse salt. Grill until a meat thermometer inserted in centers reads 168 degrees, about 5 minutes a side, placing cheese on burgers in last minute. Meanwhile, grill muffins until toasted, 30 seconds. Sandwich burgers on muffins with tomato, avocado, lettuce, and mayonnaise.