Melt butter in a skillet over medium heat. Add red onion and garlic; cook and stir until the onion softens and begins to turn brown, about 8 minutes. Scrape onion into a large bowl. Add yogurt, salt, black pepper, cayenne pepper, mustard powder, garlic powder, horseradish, parsley, cilantro, Worcestershire sauce, soy sauce, and hot sauce. Whisk together until smooth. Mix in the ground beef until evenly blended with the yogurt mixture. Cover, and refrigerate at least 1 hour.
Preheat an outdoor grill for medium-high heat. Shape mixture in to 6 patties 1/2" thick, and 5 inches across.
Grill burgers until no longer pink in the center, about 6 minutes per side.