Heat a cast-iron skillet to medium-high.
Place the turkey in mixing bowl. Finely chop drained chilies and pat them dry again; add to turkey. Season turkey with chili powder, cumin, salt and pepper. Add 3 tablespoons grated onion, 2 cloves grated garlic and about 1/2 cup cilantro to bowl. Mix burger meat and form 3 large thin patties. Add patties to the hot pan and cook 8 minutes turning once; melt a slice of each kind of cheese over top in last minute of cooking time.
Mash the avocado with remaining 3 tablespoons grated onion and 1 clove grated garlic, the jalapeño, lime juice and 2 tablespoons cilantro; season with salt to taste.
Place lettuce, onion and tomato on bun bottoms, top with cheeseburgers and the bun tops slathered with guacamole.