Combine avocado, onion, salt and pepper in small bowl. Mash with fork until well mixed, then set aside.
Melt 1 teaspoon butter in 10-inch skillet over medium-high heat until sizzling.
Break 2 eggs into pan; breaking the yolks with a spatula.
Cook until set on bottom (2 to 3 minutes).
Turn eggs over and continue cooking until eggs are cooked through (2 minutes).
Remove eggs from pan; cover to keep warm.
Repeat with 1 teaspoon butter and remaining 2 eggs.
To assemble each sandwich, spread 1 tablespoon avocado mixture onto 1 side of each bread slice. Top 1 slice of bread, spread-side up, with 1 fried egg, 2 slices tomato and 1 slice of cheese. Place 1 slice of bread, spread-side down, over cheese. Butter outside slices of bread with about 1 teaspoon of butter each.
Place 2 sandwiches in same skillet and cook over medium heat, turning once, until golden brown on both sides (2 to 3 minutes).
Repeat with remaining 2 sandwiches.