For the tahini-yogurt sauce: In a bowl, whisk together the Greek yogurt, tahini, salt, lemon zest and juice and 1 tablespoon water. Cover and reserve in the refrigerator.
For the pickled turnips: In a small saucepan over medium-high heat, combine 1 1/2 cups water with the cider vinegar, brown sugar, salt, bay leaves, garlic, beets and turnips. Bring to a quick boil to dissolve the sugar. Set aside to cool, then refrigerate until ready to serve. Serve the turnips only; discard the beets, bay leaves and pickling brine.
For the garlic butter: In a medium saucepan over medium heat, add the butter and garlic and cook until fragrant, 5 to 6 minutes. Add the parsley and mix well. Set aside in a small bowl.
For the lamb kofta burger: Prepare a grill or grill pan for medium-high heat.
In a medium mixing bowl, add the ground lamb, garlic, onion, parsley, allspice, coriander, cumin, cinnamon, salt and pepper. Mix all the ingredients together to evenly distribute the spices, being careful not to overmix. Divide into six equal portions and shape into a round, tight patties, about 4 inches wide.
Place the patties on the grill, spaced 3 inches apart to prevent flare ups. Grill for 3 minutes on the first side to develop a crust. Flip, and cook on the second side until medium-well, 2 minutes. Set aside.
Brush the cut sides of the brioche buns lightly with garlic butter and toast on the grill until golden and crisp, 5 to 10 seconds. Flip, and toast the outsides until lightly toasted, another 5 to 10 seconds.
To assemble: Layer the bottom buns with a burger, some tahini sauce, pickled turnips, lettuce, tomato and onion. Cover with the top buns and serve immediately.