Heat oven to 400 degrees. Halve spaghetti squash lengthwise and scrape out seeds; drizzle cut sides with 1 tablespoon olive oil and season with 1/4 teaspoon each salt and pepper.
Bake, cut sides down, on parchment paper-lined baking sheet 45-50 minutes, or until tender. Let cool slightly. Using fork, scrape strands of squash into bowl; reserve squash shells.
In skillet, heat remaining oil over medium-high heat; cook chicken, onion and remaining salt and pepper about 5 minutes, or until chicken is browned. Stir in pesto; bring to simmer.
Cook about 5 minutes, or until chicken is cooked through. Stir in cherry tomatoes; cook about 1 minute, or until slightly softened.
Toss spaghetti squash with chicken mixture; divide among reserved squash shells. Top with mozzarella cheese; broil 1-2 minutes, or until cheese melts. Sprinkle with chopped basil.
Tip: For a more savory dish, add chopped black olives or jarred artichokes, if desired.