Heat olive oil in a skillet over medium heat. Crack eggs into skillet; cook for 1 to 2 minutes. Flip and cook egg until done, 2 to 5 minutes more. Transfer eggs to a plate and season with salt and pepper.
Spread butter onto 1 side of each bread slice.
Heat a skillet over medium heat. Lay bread slices in the hot skillet, butter-side down. Quickly layer Swiss cheese, 1 fried egg, Cheddar cheese, 1 fried egg, avocado, and remaining bread slice, butter-side up. Carefully flip sandwich to grill other side until golden brown, 3 to 4 minutes.
Remove sandwich from skillet and cut in half.
Double recipe to make two sandwiches, which would increase avocado to one full avocado vs. half. You can also use one full avocado on one sandwich if you are like me and love avocados!
Substitute sourdough bread for the Italian bread, if desired. American cheese can be used in place of the Cheddar cheese.
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