Heat the oven to 400°F. Toss the cubed squash with olive oil, 1/2 tsp salt, and 1/4 tsp pepper. Spread on a sheet pan and roast 20-25 minutes, stirring halfway, until tender and caramelized.
Meanwhile, in a large oven-safe skillet over medium heat, cook the sausage, breaking it into crumbles, until browned (7-8 minutes). Remove and set aside, leaving the drippings in the skillet.
Add the onion and sauté 3 minutes. Stir in the garlic, thyme, and red pepper flakes, and cook 1 minute until fragrant. Add the kale and cook until just wilted (2-3 minutes).
Stir the roasted squash and sausage into the skillet and toss gently to combine. Season with the remaining salt and pepper.
Sprinkle with Parmesan, then place the skillet under the broiler for 2-3 minutes, until the cheese melts and the edges crisp. Serve warm.