1. Combine the first 5 ingredients in a medium bowl, stirring with a whisk. Add tomato, onion, cilantro, and avocado; toss gently to coat.
2. Heat a large cast-iron or nonstick skillet over medium heat. Add 3/4 teaspoon oil to pan; swirl to coat. Add 1 tortilla to pan; cook 1 minute on each side or until browned. Repeat procedure 3 times with remaining 2 1/4 teaspoons oil and tortillas.
3. Place 1 tortilla on each of 4 plates. Layer each tortilla with 1/2 cup lettuce, about 1/2 cup beans, 1/2 cup chicken, 1/4 cup avocado salsa, and 1 tablespoon cheese.