Preheat oven to 400°F. Rub cut sides of squash with 1 tablespoon olive oil and season with salt and pepper. Place cut-side down on a baking sheet and roast until tender when pierced with a knife, 35 to 45 minutes depending on size. The flesh should shred easily into strands when scraped with a fork.

While squash roasts, heat remaining 1 tablespoon olive oil in a large skillet over medium-high heat. Season chicken with a pinch of salt and pepper. Cook until golden brown on all sides and cooked through, 6 to 8 minutes total. The internal temperature should reach 165°F. Transfer to a plate and set aside.

When squash is cool enough to handle, use a fork to scrape the flesh into strands, leaving about a half-inch border intact so the shells hold their shape. Transfer strands to a large bowl, leaving the hollowed shells on the baking sheet.

Add pesto, cooked chicken, and tomatoes to the squash strands. Toss until everything is evenly coated. The residual heat from the squash will warm the pesto beautifully.

Divide the mixture evenly among the squash shells. Top with mozzarella cubes. Return to the oven and bake until cheese is melted and bubbling, 8 to 10 minutes.

Let rest 5 minutes, then garnish with fresh basil and parmesan if using. Serve warm, scooping the filling and soft squash walls together.