Preheat oven to 400°F. Brush squash halves with 1 tbsp oil, season, and roast cut-side down 35–40 minutes until strands pull apart.
While squash roasts, heat remaining oil in a skillet over medium-high. Sear chicken pieces until golden and cooked through, 5–7 minutes; remove.
In same skillet, melt butter, add garlic and cook 30 seconds until fragrant. Pour in broth and simmer 2 minutes.
Reduce heat, stir in cream and Parmesan until sauce thickens, about 3–4 minutes. Return chicken to skillet, toss to coat and heat through. Adjust seasoning.
Scrape squash into strands, divide among plates, top with creamy chicken, sprinkle parsley and extra Parmesan.