Place 2 slices of cheddar inside each hamburger bun. Using hands, press to flatten. In skillet, melt butter over medium. Working in batches, cook until the buns are crispy and the cheese melts, 1 to 2 minutes per side. Spread mayo on 2 of the grilled cheese buns. Top each with crispy bacon, lettuce, sliced tomato and the remaining buns.