Preheat oven to 400°F. Roast squash cut-side down 35–40 minutes until strands pull apart.
While squash roasts, season chicken with Italian seasoning, salt, and pepper. Heat 1 tbsp oil in skillet over medium-high heat.
Cook chicken 4–6 minutes per side until cooked through; let rest 5 minutes, then slice.
Scrape squash strands into a bowl and toss with remaining 1 tbsp oil, salt, and pepper.
Plate squash, top with sliced chicken, tomatoes, and mozzarella. Sprinkle basil and red pepper flakes.
Drizzle balsamic glaze over everything and serve warm.