1 ripe (1/2 cup) avocado, peeled, seeded
2 tablespoons chopped red onion
1/4 teaspoon salt
1/8 teaspoon pepper
8 (about 4 1/2x3-inch) slices Italian bread
8 (2x1/4-inch) slices ripe tomato
Combine avocado, onion, salt and pepper in small bowl; mash with fork until well mixed. Set aside.
Melt 1 teaspoon butter in 10-inch skillet over medium-high heat until sizzling. Break 2 eggs into pan; break yolks with spatula. Cook until set on bottom (2 to 3 minutes). Turn eggs over; continue cooking until eggs are cooked through (2 minutes). Remove eggs from pan. Cover; 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 bread, spread-side up, with 1 fried egg, 2 slices tomato and 1 slice cheese. Place 1 slice bread, spread-side down, over cheese. Butter outside slices of bread with about 1 teaspoon butter each.
Place 2 sandwiches in same skillet; cook over medium heat, turning once, until golden brown on both sides (2 to 3 minutes). Repeat with remaining 2 sandwiches.
To prevent avocado from browning, prepare avocado mixture just before assembling sandwiches.