In a medium bowl, combine the ground beef, 1 egg, red sweet pepper, parsley, and bread-crumbs until well blended. Season with salt and pepper and shape into 4 burger patties.
In a large frying pan, heat 45ml olive oil on medium heat and cook the burgers, 2 minutes per side or until cooked through. Remove from the frying pan and keep warm.
Wipe out the frying pan with a paper towel and return to medium-low heat. Heat the remaining 45 ml of oil over medium-low heat and fry the remaining 4 eggs.
Spread mustard on the bottom of each bun. Add the lettuce, tomato slices, hamburger patty, and avocado, and top each with a fried egg. Cover with the top of the bun and serve.