In a large bowl, combine the lamb, beef, shallot, garlic, oregano, cumin, paprika, salt and pepper and mix until just combined (try not to overwork the meat). Form into 4 patties, shape the patties so they are about ½ inch thick. Set aside.
Make the Easy Tzatziki:
Meanwhile, in a small bowl, combine the yogurt, olive oil, garlic, dill, cucumber, lemon juice, vinegar and salt. Stir to combine and set aside until ready to serve. This keeps well in the refrigerator for up to 6 days.
Prep the Grill:
Preheat the grill over medium-high heat (350- 400 degrees F). When the grill is hot, oil the grill grates (to do this, dip a wadded paper towel in a little oil and, using tongs, wipe the oil evenly over the grate. Be careful not to use too much oil, because that's a sure fire way to start a good flare-up—a little goes a long way here.)
Grill the Burgers:
To the greased grates, gently add the burgers to the grill. Cook on one side for 3-4 minutes until browned and they easily lift off the grates. Flip and cook on the other side until the burgers are done to your liking, about 3 more minutes for Medium. Remove from the grill and let rest for 5 minutes.
Serve in the pita pockets filled with lettuce, tomato, onion, and a big drizzle of the tzatziki.