Spread one side of the bread with the butter, then sprinkle it with the Parmesan cheese, pressing it firmly into the butter.
Place a small skillet over medium heat then add the bread, cheese side down. Cook the bread until the cheese is melted, golden brown and crispy. Set the toast aside.
Add the pesto to the same skillet, spreading it around in an even layer. Crack the eggs into the skillet and cook until the whites are cooked through the the yolks reach your desired level of doneness.
Top the toast with smashed avocado, then add the pesto eggs. Garnish with a drizzle of olive oil and a sprinkle of crushed red pepper, salt, and pepper, then serve.