Prepare a grill for medium-high heat. If using a charcoal grill, build the hot coals evenly over the bottom. Heat 2 flat cast-iron skillets over direct heat.
For the burgers and onions: Combine the lamb, coriander and mint in a large bowl. Form the meat mixture into 4 equal thin half-moon shaped patties. Season the patties with salt and pepper on both sides. Add a splash of oil to the cast-irons. Cook the patties 3 to 5 minutes, allowing the meat to get a nice crust. Flip and cook for an additional 2 to 3 minutes. Remove the burgers from the heat and let rest 2 to 3 minutes before assembling the burgers.
Meanwhile, drizzle the onions with oil and season with salt and pepper. Grill, flipping once, until softened and charred, about 5 minutes.
For the tzatziki: Add the yogurt, garlic, lemon zest, cucumber, dill and mint to a large bowl and mix to combine. Season with salt and pepper.
To serve: Spread some tzatziki over each pita, add a patty to one half of each pita, top with some grilled onion and add some chopped tomato and lettuce. Fold the other half of each pita over. Serve!