Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Combine the lamb, coriander and 3 tablespoons of the mint in a large bowl. Form the mixture into 4 patties. Season with salt and pepper. Place the patties on the hot (direct heat) side of the grill, directly over the coals. Grill until nicely browned, 3 to 5 minutes per side. Move the patties to the cool (indirect heat) side of the grill.
Drizzle the onion slices with oil and season with salt and pepper. Place on the hot (direct heat) side of the grill and cook until softened and charred, 3 to 4 minutes, moving the slice over to the cool (indirect heat) side to finish cooking if needed.
For the tzatziki, combine the yogurt, cucumber, dill, lemon zest and garlic and remaining 1 tablespoon mint in a medium bowl. Season with salt and pepper.
To serve, spoon some tzatziki onto a roll, top with a burger and a slice of grilled onion and close the roll.