1. Heat 2 tbsp. oil in an 8" skillet over high heat. Add onions; cook until browned, about 6 minutes. Transfer onions to a plate; let cool. In a bowl, mix onions, lamb, pork, mustard, herbs, spices, garlic, scallions, and salt and pepper. Form meat into two 1"-thick patties; set aside.
2. Prepare a medium-hot charcoal fire or heat a gas grill to medium-high (or heat a tablespoon of canola oil in a large cast-iron skillet over medium-high heat). Grill burgers, flipping once, until browned and cooked to desired doneness, about 10 minutes for medium rare. In a small bowl, combine feta, arugula, olives, and sun-dried tomatoes; toss with the remaining oil and lemon juice and season with salt and pepper. Place each burger on the bottom half of a bun and top each with the vegetables and bun top.