Combine sugar, yeast, and ³⁄₄ cup water heated to 115˚ in a small bowl; let sit until foamy, about 10 minutes. Combine flour and salt in a bowl and make a well in the center. Add yeast mixture and stir to form a dough. Transfer dough to a lightly floured surface; knead until smooth, about 6 minutes. Transfer dough to a lightly oiled bowl and cover with plastic wrap. Let dough rest until doubled in size, about 1 hour. Punch down dough, divide into 4 portions, and roll each portion into a ball. Transfer dough balls to a floured baking sheet. Cover with a damp tea towel and let rest for 45 minutes.

Meanwhile, make the topping: In a large bowl, combine oil, tomato paste, parsley, cayenne, cumin, paprika, and cinnamon and stir vigorously with a fork. Stir in lamb, garlic, tomatoes, onions, and chiles and season with salt; set topping aside.

Put a pizza stone in bottom third of oven and heat oven to 475°. Working with one dough ball at a time, use a rolling pin to roll dough into a 10" disk. Brush off excess flour and transfer dough to a piece of parchment paper. Spoon 3–4 tbsp. topping onto dough and using your fingers, spread topping evenly to edges. Season with salt. Holding parchment paper by its edges, transfer to baking stone. Bake until dough is golden brown and topping is cooked, 6–8 minutes. Repeat with remaining dough and topping; serve warm or at room temperature.