In the work bowl of a stand mixer fitted with paddle attachment, combine yeast, milk and egg and mix until combined. Add flour and mix on low until flour is nearly incorporated. Remove paddle and scrape clean. Replace paddle with dough hook; mix on medium-high 5-10 minutes, until smooth and elastic. Add salt and olive oil after about 2 minutes of mixing.
Coat a large bowl with a little more olive oil. Transfer dough to bowl, turning over to coat the surface. Cover with plastic wrap and set in a warm place until risen and puffy and about double its original volume, 1 1/2 - 2 hours.
Heat 1 tbsp olive oil in a large skillet over medium-high heat until shimmering. Add onions, sprinkle with salt, reduce heat to medium, and cook until onions are meltingly soft. Add lamb, breaking up large chunks with the edge of a spoon, and cook until no longer pink, 4-6 minutes. Add garlic, cinnamon, cumin, cloves, pepper and sumac. Add tomatoes, sugar, and mint. Cover and simmer until thickened. Adjust seasoning to taste; it should be really well-spiced, so add more pepper and/or cumin if needed.
Preheat oven (preferably with a pizza stone) to 450°F . Coat a 12" pan with more olive oil; stretch the dough to cover the pan. Let stand 15 minutes until slightly puffed again. Spread the warm filling on top; it should not be too thick a layer. Let stand a further 15 minutes. Bake in preheated oven 15 minutes, until golden and puffed. If filling appears to be drying out, cover with a piece of oiled parchment paper after 5-7 minutes.