Place a baking sheet on the middle oven rack and preheat to 450˚ F. Toss the squash, mushrooms, olive oil and a big pinch each of salt and pepper in a large bowl. Spread out on the hot baking sheet and roast, tossing once or twice, until the vegetables are browned around the edges and tender, 20 to 25 minutes.
Meanwhile, bring a large pot of salted water to a boil. Combine the heavy cream, garlic, sage and a pinch of salt in a small saucepan. Bring to a boil, then reduce to a simmer and cook until reduced by about one-third, 5 to 7 minutes. Remove from the heat and stir in 1 teaspoon lemon juice; add more lemon juice to taste.
Add the tortellini to the boiling water and cook as the label directs. Reserve 1 cup cooking water, then drain (reserve the pot).
Strain the cream mixture through a fine-mesh sieve into the reserved pot. Add the tortellini and 1/2 cup reserved cooking water; toss over medium heat until coated, adding a few more tablespoons cooking water as needed to loosen.
Divide the roasted vegetables among shallow bowls. Spoon the tortellini and sauce on top. Sprinkle with Parmesan and chives.