Cook the slice of bacon if needed. Let the bacon cool, and then snap (or cut) in half.
Melt 1 tablespoon butter in a small skillet over medium heat. Add the top of the hamburger bun and toast until a light golden brown. Add the bottom half of the bun and toast. Set aside.
Melt the other tablespoon butter in the same skillet. Crack the egg into the pan. Cook until the white is firm, but yolk is still runny.
Once the egg is cooked, immediately place onto the bottom half of the bun, like you would a hamburger.
Top with the bacon, a sprinkling of salt and pepper, the basil leaves, and freshly grated parmesan.
Cut in half and eat immediately.