Preheat oven to 350°.
Place squash on a baking sheet. Bake at 350° for 30 minutes or until tender; cool. Peel squash; cut in half lengthwise. Discard seeds and membranes; cut into 1/2-inch cubes. Increase oven temperature to 400°.
Bring broth, water, and sage to a simmer in a medium saucepan (do not boil). Heat oil in a large nonstick skillet over medium-high heat. Add onion; sauté 6 minutes. Add garlic; sauté 2 minutes. Add rice; sauté 1 minute. Stir in squash, broth mixture, wine, chopped thyme, salt, and pepper; cook 5 minutes, stirring occasionally.
Place rice mixture in a 13 x 9-inch baking dish coated with cooking spray. Bake at 400° for 30 minutes. Stir mixture gently. Sprinkle with cheese; bake an additional 5 minutes or until cheese melts. Garnish with fresh thyme sprigs, if desired.
Bake, peel, and cube the squash up to two days ahead.