Preheat oven to 350 degrees F
Brush 2 pie plates with olive oil, line with parchment and brush parchment with olive oil.
Cut acorn squashes in half around the center. Slice off the pointy ends so the squash halves can sit upright on a pan. Scrape out the seeds, place the squash halves on a shallow pan, drizzle with 3 tablespoons olive oil and season with salt and pepper.
Roast in oven until flesh is soft and golden, about 45 minutes. Remove and cool slightly. Scrape the flesh into a large bowl, and using a potato masher or fork, mash the squash.
While the squash is roasting, heat 1 tablespoon olive oil in a skillet over medium low. Add the shallots and garlic and sauté until soft. Add the lamb and sauté until it just turns brown. Add the cinnamon, oregano, allspice and salt and pepper and stir.
Remove the lamb from the heat and pour into a strainer over a large bowl to drain away all the fat. Set lamb aside.
In the same skillet, heat 1 tablespoon olive oil over medium low heat and add the leeks. Sauté until soft. Add the leeks, mint, walnuts, feta and dates to the squash mash and stir well. Add the beaten egg, salt and pepper to taste and stir well.
To assemble the spiral-
Unroll the phyllo dough and cover with barely damp kitchen towel to keep it from drying out while you work. Lay out a large piece of parchment as a work surface.
Place one sheet of phyllo horizontally on the parchment and brush lightly with olive oil. Place another sheet on top and brush lightly with olive oil. Fold both sheets of phyllo in half horizontally, with the fold on the bottom. Brush the folded sheet lightly with olive oil.
Spread a thin line of squash mixture (about 3 tablespoons) along the folded phyllo, just below the center. Spread a thin line of lamb (about 1 heaping tablespoon) on top of the squash mixture. Fold the ends over and then roll the phyllo up - sushi style - into an approximately 1” thick roll. Place the roll, seam side down along the edge of the pie plate. Continue to wrap the filling and coil the rolls in a spiral toward the center of the pie plate. You will probably have enough filling and phyllo for two pies.
Brush the coils with olive oil and bake at 350 degrees for about 50-60 minutes, until pastry is golden brown and crispy. Serve hot and crispy.