Prepare a grill for high heat (450° to 550°; you can hold your hand 5 in. above cooking grate only 2 to 4 seconds). Rub 1 tbsp. oil over chiles. Cook chiles, turning often, until skins are browned. Put chiles in a bowl, cover, and let cool. Seed chiles and rub off skins.
Meanwhile, rub remaining 1 tbsp. oil over chicken. Combine cumin, cayenne, and salt and sprinkle evenly over each breast. Grill chicken, turning once, until browned and cooked through, about 10 minutes total.
Squeeze a little lime over chicken, then top each breast with a few avocado slices, a reserved chile, and about 1/4 cup cheese. Grill, covered, until cheese melts, about 4 minutes.
Note: Nutritional analysis is per serving.