Halve, seed, and peel avocado. Place one avocado half in a small bowl; mash with a fork or the back of a wooden spoon. Stir in mayonnaise, lemon juice, and garlic. Thinly slice the remaining avocado half. Set aside.
In a large skillet melt butter over medium heat. (Or coat a large skillet with cooking spray.) Break eggs into skillet. If desired, sprinkle with salt and pepper. Reduce heat to low; cook eggs for 3 to 4 minutes or until whites are completely set and yolks start to thicken.
For fried eggs over easy or over hard, when the whites are completely set and the yolks start to thicken, turn the eggs and cook for 30 seconds more (over easy) or 1 minute more (over hard).
Layer four of the bread slices with cheese (if using), avocado slices, bacon, and fried eggs. Spread the remaining four bread slices with mayonnaise mixture; place on sandwiches, spread sides down.