Using a 2 1/2-inch round cookie cutter, cut a hole from the center of 4 bread slices. Discard bread rounds or reserve for another use.
Heat a large skillet over medium. Add bacon; cook 6 minutes. Increase heat to medium-high; cook 2 minutes or until crisp. Place bacon on a paper towel-lined plate. Pour bacon drippings into a bowl and reserve.
Combine avocado, mayonnaise, and salt in a small bowl; mash to combine. Stir in bacon.
Add half of reserved bacon drippings to pan over medium-high. Place 2 cut bread slices in skillet; break 1 egg into each hole. Sprinkle 1/8 teaspoon pepper on each egg. Cook 2 minutes or until eggs begin to set. Carefully turn bread; cook 2 minutes or until eggs are set. Remove from pan. Repeat procedure with remaining half of bacon drippings, cut bread slices, eggs, and pepper.
Spread avocado mixture evenly over 4 uncut bread slices. Top each with 2 lettuce leaves, 2 tomato slices, and 1 egg-in-a-nest bread slice.