Preheat the grill. Preheat your grill for direct grilling at medium-high heat (around 350-400 degrees F).
Prepare your condiments. To make the tzatziki sauce, place the grated cucumber in a paper towel and press to remove moisture. Combine the dried cucumber with Greek yogurt, lemon juice, salt, grated garlic, onion powder, and white pepper. Stir and then refrigerate until ready to serve. To make the mint sauce, combine the mint, parsley, lemon juice, garlic, salt, pepper, red pepper flakes, and olive oil in a blender with a tight-fitting lid. Blend until there are no large pieces of garlic or herbs remaining, but not quite a totally smooth puree.
Make your lamb burger patties. In a large mixing bowl, combine the ground lamb, breadcrumbs, eggs, curry powder, salt, and pepper. Mix gently with your hands until the seasoning is well distributed through the meat and then form 6, 1/3 pound, patties.
Grill the burgers. Place the lamb burger patties directly on the grill grates, close the lid, and grill for about 4-5 minutes. Open the lid and flip the patties, being careful not to press or squish them with your spatula. Close the lid and continue cooking until the lamb reads 155 degrees F. Transfer the lamb burgers to a plate and cover lightly with foil so the carry-over cooking will take the lamb up to 160 degrees F.
Assemble your burgers. While the lamb rests for 5 minutes, toast your buns on the hot grill and gather up your tzatziki sauce, mint sauce, and other burger toppings. Place the burgers on the buns and then top with your desired toppings. Spoon a generous serving of tzatziki over the burgers and spread a spoonful of the mint sauce on the top bun. Take a giant bite and enjoy!