Make the dough: In a small bowl, mix the yeast, sugar and 1/3 cup (79 ml) lukewarm water. Set aside until it starts to foam, about 15 minutes.
In the bowl of a stand mixer fitted with the dough hook, sift together the flour and salt. (If making the dough by hand, mix in a medium bowl and make a well in the center.)
Pour the yeast mixture into the flour and add the remaining 2/3 cup (158 ml) warm water. Set mixer to medium speed and incorporate the liquid into the flour until most of the liquid is absorbed and a somewhat shaggy dough is formed. Add a bit of extra water if the dough seems too dry, or a small amount of extra flour if it is too sticky.
Continue to knead the dough for 7 to 10 minutes until it becomes elastic and smooth. If kneading by hand, turn the dough onto a floured board and make one-quarter turns at regular intervals.
Coat a large bowl with vegetable oil and place the dough in the bowl. Cover with a damp kitchen towel and leave in a warm place for 1 hour or until the dough doubles in size.
Heat the oven to 425˚F (218˚C). If using a pizza stone, heat the stone starting in a cold oven, or line a baking sheet with parchment and set aside.
Make the topping: Melt the butter and 1 tablespoon olive oil in a medium skillet over moderate heat. Add the onions, garlic and 1/4 teaspoon salt. Sauté until the onions are soft and translucent, but not brown, about 6-8 minutes.
In a medium bowl, combine the lamb, tomato paste, Aleppo pepper, oregano, mint, sumac, the remaining 1/2 teaspoon salt and black pepper. Add the sautéed onions and garlic.
When the dough has doubled in size, turn it out onto a lightly floured surface. Cut the dough into quarters. With floured fingers, shape the dough into four 1/4 inch (6 mm) thick round or oblong pizzas. If necessary, use a rolling pin to flatten each piece.
Place one flatbread on the parchment-lined baking sheet or on a pizza peel, lightly coated with flour. Brush the edges of the dough with some of the remaining olive oil to create a crisp crust. Spread a thin layer of the topping mixture evenly over the dough. Place the baking sheet into the oven or slide the flatbread from the peel onto the pizza. Bake for about 15 minutes until the edges begin to turn golden and the lamb topping is browned.
Remove the lahmacun from the oven and slide onto a cutting board. Cut into 4-6 wedges. Repeat with remaining dough and toppings to make three more pizzas. Serve immediately with lemon wedges, parsley and fresh tomato slices.