Heat butter in a skillet over medium heat and cook and stir red onion until soft and translucent, about 5 minutes. Mix ground beef, cooked onion, mushrooms, ketchup, egg yolks, Worcestershire sauce, horseradish mustard, black pepper, and sea salt in a large bowl until thoroughly combined.

Form the mixture into six 3/4-inch thick patties and cook in a large skillet over medium heat until burgers are juicy and browned, 5 to 7 minutes per side. An instant-read meat thermometer inserted into the center of a patty should read at least 160 degrees F (70 degrees C).