To make the pickled onions: Place the onion, lime juice, salt, and sugar in a small bowl. Stir to combine, cover, and let sit at room temperature for about 2 hours to soften. Refrigerate until ready to use.
To make the herbed yogurt sauce: In a small bowl, stir together the yogurt, lemon juice, garlic, herbs, and 1/2 tsp salt. Adjust the salt to taste. Cover and refrigerate until ready to serve, or for up to 2 days.
To make the lamb burgers: In a small skillet over medium heat, warm the olive oil. Add the onion and cook, stirring frequently, until softened, about 7 minutes. Transfer to a small plate to cool.
In a large bowl, combine the lamb, pork, mint, dill, parsley, garlic, cumin, coriander, salt, pepper, and cooled onions. Gently mix with your hands. Do not overwork the meat. Divide the mixture into six equal balls. Press into patties and transfer to a parchment-lined baking sheet. (If not cooking immediately, cover and refrigerate for up to 8 hours.)
Heat a cast-iron skillet over medium-high heat until just smoking. Working in batches, sear the burgers until well browned, 2 to 3 minutes per side for medium-rare, or about 5 minutes per side for well-done. Alternatively, prepare a grill for cooking over medium-high heat. Lightly grease the grill grate. Place the burgers on the grill, and cook, turning once, to reach desired doneness, about 3 minutes per side for medium-rare, or about 5 minutes per side for well-done.
Transfer the burgers to a plate to rest for 5 minutes before serving. Top with a generous dollop of herbed yogurt sauce and some pickled onions. Add greens and sliced tomatoes or cucumbers. Serve immediately.