Preheat oven to 400 degrees F.
In a large skillet over medium-high heat, add 2 tsp. oil, diced chicken, ½ tsp. poultry seasoning and ¼ tsp. salt. Cook chicken until lightly browned, about 2 minutes. Add the onions; toss with chicken and cook another 3 minutes until onions are softened.
Remove from heat and mix in mashed squash, nutmeg and apple slices.
Brush a rectangular baking sheet with oil and spread pizza dough to make a 10-inch x 15-inch rectangle. On the longer sides cut slits in dough, keeping attached to the center, about 1 1/2 inches apart and 3 inches deep. Sprinkle half of shredded cheese down the center. Evenly spread squash and chicken mixture. Top with remaining cheese.
Braid dough by lifting strips across filling to meet on the opposite side at a slight angle. Continue alternating strips. Fold at ends to seal dough. Lightly brush with remaining 2 tsp. olive oil and sprinkle with remaining poultry seasoning.
Bake 20-25 minutes or until deep golden brown. Remove from oven and let cool 10 minutes. Slice and serve.