Make the dough: Combine the yeast, sugar, olive oil and 2 cups warm water; let stand 5 minutes. Whisk both flours and the salt in a large bowl. Add the yeast mixture to the flour mixture and stir with a fork until the dough forms. Turn out onto a lightly floured surface; knead until smooth and elastic, about 6 minutes. Place in a bowl brushed with oil, cover and let stand until doubled in size, about 1 hour.
Punch down the dough and divide into 12 pieces. Roll each piece into a ball, cover and let stand until doubled in size, about 45 minutes.
Meanwhile, make the topping: Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the pine nuts, onion and bell pepper; cook, stirring, until softened, about 6 minutes. Stir in the tomato paste and cook 1 minute. Add the remaining 1/4 cup olive oil, the lamb, cumin, cinnamon, red pepper flakes, allspice and 1/2 teaspoon salt. Cook, breaking up the meat, until browned, 5 minutes. Stir in the tomatoes and cook 1 minute. Remove from the heat and stir in the yogurt, lemon juice, cilantro and parsley; season with salt.
Make the sauce: Stir the yogurt, mint, lemon juice and 1/4 cup water in a bowl. Season with salt and pepper. Refrigerate, covered, until ready to use.
Preheat the grill to medium. Cut twelve 9-inch squares of parchment paper. Stretch each piece of dough into an 8-inch round, brush with olive oil on both sides and stack the rounds, separating each with parchment.
Invert the dough onto the grill; remove the paper. Grill in batches until golden brown on the bottom, 1 to 2 minutes. Flip and spread each round with 1/4 cup lamb mixture, leaving a border. Continue grilling until the dough is charred in spots and cooked through, about 3 minutes. Transfer the pizzas to a platter. Drizzle with olive oil and serve with the sauce.