1. Make the flatbreads: Combine yeast and water in a mixer bowl. Let stand until foamy, about 5 minutes. Add oil, sugar, and salt. Mix with the dough hook on medium speed until well combined. Reduce speed to low. With the machine running, add flour, a little at a time, and mix until dough comes together. (You will need more flour in warm weather than in cold.) Transfer dough to a lightly floured work surface, and knead until smooth and supple, 10 to 15 minutes. Transfer to an oiled bowl, and cover with plastic wrap. Refrigerate for at least 4 hours (or overnight).
2. Remove dough from refrigerator, and bring to room temperature. Place a pizza stone or baking sheet on the floor of the oven. Preheat oven to 500 degrees.
3. Make the topping: Melt butter with oil in a medium skillet over medium heat. Add leeks and scallions; season with salt and pepper. Cook, stirring occasionally, until softened, about 5 minutes. Transfer to a bowl. Stir in parsley and mint.
4. Add lamb to the same skillet, and set over medium-high heat. Cook, stirring and breaking meat into small pieces, until browned, 6 to 7 minutes.
5. Divide dough into 4 portions. On a lightly floured surface, gently stretch 2 pieces into 14-by-7-inch rounded rectangles; keep other portions covered.
6. Place dough rectangles on a parchment-lined peel or inverted rimmed baking sheet. Brush with oil, and spread each with 1/4 of the leek mixture, leaving a 1/2-inch border. Top each with 1/4 of the lamb, then sprinkle with 1/4 cup feta. Season with salt and pepper.
7. Carefully slide parchment with flatbreads off the peel or sheet onto the pizza stone. Bake until flatbreads are golden brown, 10 to 12 minutes. Repeat with remaining dough.
8. Garnish flatbreads with fresh mint, and drizzle with oil. Cut into thick slices, and serve immediately.

Serves 8-10.