Preheat oven to 425°F with the rack in the middle position. Line a large, rimmed baking sheet with heavy gauge aluminum foil and rub 1 teaspoon of olive oil all over. The olive oil layer will be exceptionally thin.
Place a large pot of water over high heat and bring to a boil. Reduce to medium heat and add the whole raw sausages. Par-boil for 5 minutes. Remove from the water and set aside.
Trim squash ends and cut the squash crosswise into 4 equal rounds. Scoop out the seeds and discard. Place the rounds and the ends on the prepared baking sheet and drizzle with 1 tablespoon of oil and sprinkle with 1/4 teaspoon pepper and 1/8 teaspoon salt. Use your fingers to distribute the oil, pepper, and salt evenly. Turn the ends cut side down.
Place the baking sheet into the preheated oven for 25 minutes and bake the rounds and ends UNDISTURBED.
While squash is baking, combine mushrooms, tomatoes, and the remaining 1 tablespoon oil, 1/4 teaspoon pepper and 1/8 teaspoon salt in a large bowl.
At the end of the 25 minutes, carefully remove the baking sheet from the oven. Turn the ends over, exposing the flesh, and do the same with the squash circles. There should be small amounts of golden-brown patches over the surfaces. This will increase as the bake continues. Scatter the tomato and mushroom mixture among the half-baked squash circles and ends. Lay the sausages around the mix place all back into the hot oven for another 20 minutes. Turn the sausages half-way through to promote browning on both sides.
Remove pan from oven. Use a fork to shred the squash ends first, then the centers of the rings into long strands. Divide the well-caramelized strands from the ends among the four circles and fluff all together within each circle has an equal mix of interior and exterior threads.
Next slice each sausage link into thick chunks on the diagonal and divide along with the mushrooms and the tomatoes and 1/8 cup freshly grated Parmesan cheese among the fluffed nests. Turn the oven to the broil setting and place the nests on the rack (still in middle position) for 1 to 2 minutes.
Use a wide metal spatula to carefully remove each nest to a plate. Garnish with the remaining 1/8 cup freshly grated Parmesan cheese, chopped and whole fresh basil leaves, and a healthy drizzle of Balsamic glaze. Offer a simple side dish of fresh garlic bread or a salad and heaps more fresh Parmesan for passing at the table.