Remove white papery skin from garlic heads (do not peel or separate the cloves). Coat each head with cooking spray; wrap separately in foil. Bake at 350° for 1 hour; let cool 10 minutes. Separate cloves; carefully snip or cut 1 end of each clove. Squeeze cloves over a small bowl to extract garlic pulp. Discard skins.
Sprinkle Parmesan cheese over garlic pulp; mash with a fork until blended. Stir in mayonnaise until blended. Cover tightly with plastic wrap pressed onto surface of aioli; chill.
Place chicken breasts between 2 sheets of heavy-duty plastic wrap or wax paper; flatten to 1/4-inch thickness, using a meat mallet or rolling pin (if chicken breasts are very large, cut in half, crosswise). Sprinkle salt and pepper over 1 side of each chicken breast.
Place a large nonstick skillet over medium-high heat until hot. Coat chicken breasts with cooking spray. Add chicken breasts, seasoned side down; cook 2 minutes on each side or until done. Remove from pan, and keep warm. Coat cut sides of buns with cooking spray; place in skillet, cut sides down, and cook until lightly toasted.
Spread 1 tablespoon garlic aioli over toasted sides of each bun. Divide chicken evenly among bottom halves of buns; top with lettuce, tomato, and top halves of buns. Serve immediately.