Warm vegetable oil in a large skillet over medium-high heat. Add chopped onion and cook until softened, about 3 minutes. Add garlic and cook, stirring, 1 minute. Stir in chicken and 1/4 tsp. salt and cook until heated through, 1 to 2 minutes.
Place flour tortillas between 2 moistened paper towels and microwave on high for 10 seconds to soften. Spoon some shredded lettuce onto each tortilla. Top each with chicken, salsa and shredded Cheddar. Roll up burritos, cut in half and serve.