Melt butter in nonstick skillet set over medium heat. Break eggs into skillet; cook for 2 to 3 minutes until egg whites are just set for sunny-side up or cook until desired doneness. Season with salt and pepper.
Stir together pesto and mayonnaise. Spread over toasted buns. Layer cheese, arugula, tomato slices and fried eggs over bottom halves of buns. Cap with top halves of buns and avocado.
Substitute bagels, English muffins or burger buns for ciabatta buns if desired.