For sauce: In a small bowl, stir together yogurt and cucumber. Cover and chill until serving time.
In a medium bowl, combine onion, jalapeño pepper, mint, cumin, garlic, and a pinch of salt. Add ground beef; mix well. Form beef mixture into two 3/4-inch-thick patties.
For a charcoal grill, place patties on the rack directly over medium coals. Grill, uncovered, for 14 to 18 minutes or until patties are done (160F), turning once. If desired, serve patties on flat bread. Spoon cucumber sauce over patties.