Step 1

FOR THE FILLING: Heat oil in 10-inch skillet over medium-high heat until shimmering. Add onion and salt and cook until softened, about 5 minutes. Stir in garlic, ginger, and pepper and cook until fragrant, about 30 seconds. Stir in wine, bring to simmer, and cook, stirring occasionally, until onion is very tender and mixture has reduced slightly and measures 1¼ cups, 15 to 20 minutes.

Step 2

Transfer onion mixture to food processor and let cool slightly. Add pumpkin and eggs and process until mixture is thoroughly combined and smooth, about 3 minutes, scraping down sides of bowl as needed. Combine halloumi, cottage cheese, and mint in separate bowl.

Step 3

FOR THE LAYERS: Adjust oven rack to middle position and heat oven to 400 degrees. Whisk milk and egg together in bowl until combined. Trim 50 phyllo sheets to 12½ by 8½ inches.

Step 4

Spread 1 cup pumpkin filling on bottom of greased 13 by 9-inch baking dish. Lay 5 phyllo sheets in dish, brush with egg mixture, then top with 5 more phyllo sheets.

Step 5

Brush phyllo with egg mixture, then spread 2⅓ cups pumpkin filling evenly on top. Lay 5 more phyllo sheets in dish, brush with egg mixture, then top with 5 more phyllo sheets.

Step 6

Brush phyllo with egg mixture, then spread half of cheese mixture evenly on top. Lay 5 more phyllo sheets in dish, brush with egg mixture, then top with 5 more phyllo sheets.

Step 7

Working from center outward, use palms of your hands to gently compress layers and press out any air pockets. Repeat layering in steps 5 and 6, then brush top with egg mixture.

Step 8

Trim 10 phyllo sheets to 13 by 9 inches. Spread remaining pumpkin filling evenly on phyllo layers. Lay 5 large phyllo sheets in dish, brush with egg mixture, then top with 5 more large phyllo sheets. Gently compress layers and wipe away excess filling that may have leaked out along sides of dish. Brush top with egg mixture and bake until borek registers 165 degrees and top is puffed and golden brown, 40 to 45 minutes. Let cool for 30 minutes before serving.