Peel and finely chop the onion. Wash parsley, shake dry. Pluck off leaves and chop finely. Mix egg yolk, vinegar, 1 tsp. mustard, salt and pepper in a bowl with the whisk of the hand mixer for about 2 minutes.
Add 8 tablespoons of oil, first drop by drop, then in a thin stream, stirring continuously. Stir in half the diced onion and half the parsley.
Knead minced meat, 1 tablespoon mustard, breadcrumbs, remaining onion cubes and remaining parsley, season with salt and pepper. Shape into 4 patties. Heat 2 tablespoons of oil in a frying pan and fry the patties for 8-10 minutes while turning.
Peel the red onion and cut into rings. Wash and clean the salad and shake dry. Wash, clean and slice the tomatoes.
Toast the burger buns and cut in half horizontally. Layer lettuce, mayonnaise, patty, cheese, ketchup, onion rings and tomatoes on the lower halves of the buns. Place the upper halves of the rolls on top. Use the rest of the mayonnaise for other purposes.