Melt 2 teaspoons butter in a skillet over medium heat; add the eggs. Allow the egg whites to cook until mostly firm before breaking the yolks; continue cooking until eggs are completely cooked and no longer runny, 2 to 3 minutes.
Spread one side of toasted sourdough bread slice with Dijon mustard.
Arrange avocado slice onto the mustard.
Top avocado with cooked eggs.
Sprinkle Parmesan cheese over eggs.
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