Cook the Filling: Warm the olive oil in a large skillet over medium heat. Add the chopped leeks and carrots, cooking for 8-10 minutes until softened but not browned. Stir in the ground lamb along with the cinnamon, black pepper, paprika, allspice, and salt. Increase the heat to medium-high and cook, stirring occasionally, until the lamb is fully browned, about 10-15 minutes. Remove from heat, allow the mixture to cool completely, and drain off any excess fat.
Prepare the Puff Pastry: While the lamb mixture cools, divide the puff pastry in half if it's in a single sheet. On a floured surface, roll each piece into a rectangle about 9 × 11 inches in size. Line the bottom and sides of an 8 × 11-inch baking dish (or a 2-quart gratin dish) with one piece of the pastry, allowing the edges to slightly overhang.
Assemble the Pie: In a mixing bowl, whisk two of the eggs. Add the cooled lamb mixture, along with the olives, parsley, mint, and dill. Stir everything together until well combined. Transfer this mixture into the pastry-lined baking dish, spreading it evenly. Cover with the second sheet of puff pastry, folding and pinching the edges to seal the pie completely. Chill the assembled pie in the refrigerator for at least 1 hour (or up to 4 hours), or place it in the freezer for 15 minutes to firm up.
Bake the Pie: Preheat your oven to 400°F. Whisk the remaining egg and brush it over the top of the pastry to create a golden finish. Sprinkle the sesame seeds evenly over the top. Bake the pie for about 35 minutes, or until the pastry is puffed and golden brown. The surface may appear slightly wet when it first comes out of the oven, but the pastry will absorb the moisture as it cools.
Cool and Serve: Allow the pie to rest for 15-20 minutes to cool slightly before cutting it into squares. Serve warm, optionally accompanied by garlicky yogurt sauce. Leftovers can be reheated in a 350°F oven for best results.