Prepare a grill for medium-high heat; thoroughly clean grates and brush with oil. Whisk yogurt and lemon juice in a medium bowl; season with salt and pepper. Set aside until ready to assemble.
Mix garlic, tomato paste, cinnamon, cumin, paprika, and 2 1/2 teaspoons salt in a large bowl. Mix in beef and 2 tablespoons olive oil until completely combined. Heat remaining 2 tablespoons oil in a large cast iron skillet on the grill or stove over medium high. Cook beef mixture, stirring occasionally and breaking up large clumps, until browned and cooked through, 12-14 minutes. Add onion and cook until just beginning to soften, about 2 minutes; remove from heat and set aside until ready to assemble.
Increase grill to high heat. On a clean work surface lightly dusted with flour, stretch or roll each piece of dough into a 12- x 10-inch oval and transfer to a pizza peel or cookie sheet dusted with cornmeal. (If dough springs back, cover and let rest 10 minutes, then stretch again, resting as needed). Transfer both pieces of prepared dough to grill over direct heat. Cook until first side is charred and dough is dry and stiff, about 2 minutes. Turn and cook just until second side is charred and cooked through, about 2 minutes. Transfer to baking sheet and let cool slightly. Top with cooked beef mixture, dividing evenly. Dollop with yogurt mixture and top with cucumber and cilantro.