Whisk half-and-half, 1 egg, sugar, cinnamon, and salt together in a bowl until batter is smooth.
Place 2 bread slices on a work surface and top each with 1 slice American cheese, 1 slice Cheddar cheese, and crumbled bacon. Top each with a second piece of bread, making a sandwich. Cut a 2-inch hole in the center of each sandwich using a biscuit cutter or 2-inch wide glass.
Melt butter in a large nonstick skillet over medium-low heat, tilting the skillet to cover entire surface with butter. Dip both sandwich rings and cut-out holes in the batter until evenly coated.
Place the dipped sandwich rings and cut-out holes in the melted butter in the skillet. Immediately crack an egg into the hole of each ring. Cook until egg white is set, bread is golden brown, and cheese is softly melted, about 2 minutes per side.