Prepare the Pickled Onions: In a small bowl, combine the red onion, lime juice, kosher salt, and cane sugar. Stir to combine and let sit at room temperature for about 2 hours. Refrigerate until ready to use.
Make the Herbed Yogurt Sauce: In another small bowl, mix the Greek yogurt, lemon juice, garlic, and chopped herbs. Add salt to taste. Cover and refrigerate until ready to serve, up to 2 days.
Cook the Onion for the Burgers: Heat olive oil in a small skillet over medium heat. Add the diced onion and cook, stirring frequently, until softened (about 7 minutes). Transfer to a plate to cool.
Prepare the Lamb Burger Mixture: In a large bowl, combine the lamb, pork, mint, dill, parsley, garlic, cooled onions, cumin, coriander, salt, and pepper. Mix gently with your hands, being careful not to overwork the meat. Shape the mixture into six equal patties. Place patties on a parchment-lined baking sheet and refrigerate for up to 8 hours if not cooking immediately.
Cook the Burgers: Heat a cast-iron skillet or grill over medium-high heat until hot. Cook the burgers for about 3 minutes per side for medium-rare, or 5 minutes per side for well-done.
Assemble and Serve: Let the burgers rest for 5 minutes before serving. Top with a dollop of herbed yogurt sauce and pickled onions. Serve over mixed greens or in buns, with optional tomatoes and cucumbers on the side.