Bring a large pot of water to a boil. Preheat oven to 425°F. Place all the diced butternut squash onto a baking sheet, drizzle with 2 tablespoons olive oil, the ground sage, 1 tsp poultry seasoning, salt and pepper. Toss the squash with the oil and seasonings and roast for 30 minutes or until fully softened and beginning to caramelize. Reduce oven heat to 350°F
Once the pot of water comes to a boil, add a generous pinch of salt along with the lasagna noodles. Cook until al dente, remove from heat, rinse under cold water to stop the cooking process, and set aside.
In a medium skillet on medium heat, add 1 tablespoon olive oil (if using raw chicken, you may need a bit more) and the shredded chicken or turkey. Add 1 teaspoon poultry seasoning along with the paprika, salt and pepper. Combine and mix until just warmed through. Remove from skillet and set aside. Add in the last tablespoon of olive oil along with the mushrooms, salt and pepper, thyme and last teaspoon of poultry seasoning. Cook until the mushrooms have released all their liquid and the liquid has evaporated. Remove from skillet and set aside.
Add the butter to the skillet along with the garlic, cook for about 30 seconds or until the butter is melted. Add in the flour and whisk until well combined with the butter and cooked through. Add in the milk in 2 to 3 parts whisking in between until a slightly thick consistency is achieved. When you sweep a wooden spoon through the sauce in the skillet, you should be able to see the bottom of the pan for about 1 second. The sauce will continue to thicken while cooling. Prepare a 9×13 baking dish and place 2 ladle-fulls of the bechamel on the bottom.
Place a lasagna noodle in front of you. Cover the top with 1 teaspoon of the bechamel, add about 1 teaspoon turkey/chicken followed by a small amount of butternut squash and mushrooms. Roll up carefully and place in the baking dish seam side down. Continue until all noodles are used up. At this point, if the bechamel is too thick, add ¼ cup of milk at a time until desired consistency is reached. Pour the remainder over all the rolls and top with a handful of fresh grated Parmigiano Reggiano. Bake at 350°F for about 15 minutes, the broil for 5 minutes or until a golden crust is achieved. Enjoy!