Place 2 strips of bacon in a skillet and cook over medium heat until browned, about 7 minutes. Set bacon slices on paper towel.

Whisk 3 eggs with salt and pepper in a small bowl until frothy.

Melt 2 teaspoons butter in skillet over medium heat. Pour in beaten eggs; let cook for 30 seconds. Lift the edges of the omelet so that the uncooked egg runs under the cooked edges and comes into contact with the hot skillet. Shake and tilt the skillet to move the uncooked egg. Repeat until the top is set, about 3 minutes; reduce heat to low.

Chop bacon roughly; sprinkle over omelet. Sprinkle with cheese; allow cheese to melt, 1 to 2 minutes.

Butter 2 slices of bread with 1/2 teaspoon butter each. Lay bread, buttered side up, side-by-side over omelet, leaving about 1 inch between slices. The melted cheese will help the bread adhere.

Run a spatula around the entire omelet to loosen from the skillet. Carefully turn the omelet over so buttered side of bread is in contact with the skillet. Cook until bread is toasted and golden, about 2 minutes.

Using the spatula, fold in edges of omelet to align with sides of the bread. Cut omelet along center with the spatula to form 2 sandwiches. Butter 2 remaining bread slices with remaining butter; place butter side up on each sandwich. Turn sandwiches; cook until golden, about 2 minutes more. Serve and enjoy.