DO-AHEAD TIP: Chill burger mixture for at least 1 hour (see recipe).
Melt the butter in a skillet over medium heat; add red onion and garlic; cook and stir until onion softens and begins to turn brown, about 8 minutes; transfer to a large bowl.
Add remaining ingredients except ground beef and burger buns; whisk until smooth. Add ground beef and blend well with yogurt mixture. Cover and refrigerate for at least 1 hour.
Preheat grill to MEDIUM-HIGH. Shape beef mixture into 6 patties, 1/2-inch thick and 5 inches in diameter; grill for 6 minutes per side or until no longer pink in the center.
Serve on burger buns with your favorite condiments.