Prep the squash. Heat the oven to 400°F (200°C).
Carefully slice the spaghetti squash in half lengthwise and scoop out the seeds. Rub the cut sides with 1 tablespoon of olive oil and season with salt and pepper. Place cut side down on a parchment-lined baking sheet.
Roast until tender. Bake for 35–45 minutes, depending on size, until the flesh is easily pierced with a fork.
Let it cool slightly, then use a fork to scrape the flesh into strands. Pat dry with paper towels if it looks watery. Reduce oven to 375°F (190°C).
Sauté the aromatics. In a large skillet, heat 2 tablespoons olive oil over medium heat.
Add the onion and cook until soft and translucent, about 5 minutes. Stir in the garlic and cook 30–60 seconds until fragrant.
Build the sauce. Pour in the chicken broth and bring to a gentle simmer. Stir in the half-and-half, 1 cup of the Parmesan, Italian seasoning, and red pepper flakes.
Simmer 2–3 minutes until lightly thickened. Season with salt and pepper. The sauce should coat the back of a spoon.
Combine with chicken and squash. Add the shredded chicken, spaghetti squash strands, and parsley to the skillet.
Toss until everything is evenly coated. Taste and adjust salt, pepper, or red pepper flakes as needed.
Assemble the bake. Transfer the mixture to a lightly greased 9x13-inch baking dish. Top with mozzarella and the remaining 1/2 cup Parmesan, spreading evenly.
Bake to bubbly perfection. Bake at 375°F for 18–22 minutes, until the cheese is melted and the edges are lightly golden.
For extra color, broil for 1–2 minutes at the end, watching closely.
Finish and serve. Let it rest 5–10 minutes so it sets slightly. Sprinkle with extra parsley and a crack of black pepper. Serve warm.