Preheat oven to 375 degrees. Using a sharp knife, split each chicken breast in half horizontally, then pound between two pieces of plastic wrap until about 3/8 inch thick (you should have 8 flattened pieces). Season on both sides with salt and pepper.
Heat oil in a large skillet over medium-high heat. Working in batches, saute chicken breasts and garlic until chicken is light brown on one side only, about 2 minutes; discard garlic.
Transfer chicken to a baking sheet, browned side down. Sprinkle chicken pieces with half the cheese; layer each with a tomato slice and 4 asparagus pieces. Roll up; secure with a toothpick. Sprinkle remaining cheese over rolls; bake until chicken is cooked through, 15 to 20 minutes. Remove toothpicks. Serve hot.