Prepare and chill the pesto up to two days ahead, and bring to room temperature before serving. Bake and chill the squash halves a day or two before; it's actually easier to remove the flesh when it's cold. Reheat the cold squash in the microwave. The unique pesto would also be good on a pizza with sun-dried tomatoes.
Cut each squash in half lengthwise; discard seeds. Place squash halves, cut sides down, on a baking sheet coated with cooking spray. Bake at 350° for 1 hour or until tender. Cool slightly. Scrape inside of squash with a fork to remove spaghettilike strands to measure about 8 cups. Place in a large bowl. Sprinkle with 1/4 teaspoon salt; toss gently to combine. Cover and keep warm.
Place cilantro, broth, oil, pepper, remaining 1/4 teaspoon salt, garlic, and edamame in a food processor; pulse until coarsely chopped. Serve edamame pesto over squash; sprinkle with cheese.