Roast squash: Preheat oven to 400°F. Halve squash, remove seeds, brush with 1 tbsp oil, season with salt and pepper. Place cut side down on a baking sheet and roast until fork-tender, 35–40 minutes. Scrape strands and reserve shells.

Sear chicken: Cut chicken into bite-size pieces, season with salt, pepper, and Italian seasoning. Heat remaining 1 tbsp oil in an ovenproof skillet over medium-high. Cook chicken until golden and cooked through, 5–6 minutes. Transfer to a plate.

Build sauce: Reduce heat to medium, melt butter, cook garlic 30 seconds. Add heavy cream, simmer 2 minutes until thickened. Off heat, whisk in Parmesan and a pinch of nutmeg. Season with salt.

Combine: Add squash strands and chicken to the skillet. Toss to coat. Spoon mixture into reserved squash shells on the baking sheet.

Broil: Top with mozzarella and broil 6 inches from heat until bubbly and golden, 2–3 minutes. Garnish with parsley and serve.