Mix mayonnaise with garlic. Spread 1 tablespoon of the mixture on each of 4 slices of bread. Top each with 1 slice Edam, a few leaves of lettuce, a slice or two of the tomato, and 1/4 of the avocado slices. Set aside.
Fry bacon in cast-iron or nonstick skillet until crisp; remove to plate lined with paper towels and drain and pat dry. Remove all but 1-2 tablespoons bacon fat from skillet. Crack eggs into skillet and fry until whites are just set; flip over and cook for 1 minute. Place 2 bacon slices on top of avocados on each of the prepared bread slices. Top bacon with an egg, a slice of Cheddar, and a slice of bread. Wipe the skillet out and return to heat. Add 1 tablespoon butter and, when melted, add sandwiches to skillet. Spread sandwich tops with butter. Grill sandwiches until bottom is lightly browned and cheese begins to melt. Flip sandwiches, adding more butter if necessary. Grill until browned and cheese melts.