To prepare relish, combine tomatoes, 1 tablespoon mint, and 1 tablespoon juice; cover and chill until ready to serve.
To prepare sauce, place feta cheese and next 4 ingredients (through garlic) in a food processor, and process 2 minutes or until smooth, scraping sides. Cover and chill.
Prepare grill or broiler.
To prepare burgers, combine 1/2 cup mint and next 6 ingredients (through beef). Divide mixture into 5 equal portions, shaping each into a 1/2-inch-thick patty. Place patties on a grill rack or broiler pan coated with cooking spray; cook 5 minutes on each side or until done. Place 1 patty on bottom half of each bun; top with 1/4 cup relish. Drizzle each serving with about 2 tablespoons sauce; top with top halves of buns. Serve immediately.
Wine note: The sauce of lemon, feta cheese, and yogurt on this tasty burger adds a salty, tangy flavor that can make many red wines taste flat and dull. So opt for a fresh, clean, dry white wine instead (whoever said white wines don't go with red meat?). My favorite choice: a snappy pinot grigio, served well chilled. The 2003 Zenato Pinot Grigio from Veneto, Italy, is an affordable choice at about $10. -Karen MacNeil