Horseradish mayonnaise: Whisk the egg yolk and Dijon mustard in a bowl, gradually add the oil a little at a time, continuously stirring. Finally, add the vinegar and horseradish. Season with salt and pepper.
Prepare the hamburger patties according to the recipe.
Sauté the onion in butter over low heat, stirring occasionally, for about 15 min. Add sugar and sauté for a few more minutes.
Season the hamburger patties with salt and pepper. Fry or grill for about 5-6 min on each side.
Fry the buns in butter with the cut side down until they are colored.
Assemble the hamburgers with mayonnaise, lettuce, meat, cheese, and caramelized onions.