Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with aluminum foil.
Spray spaghetti squash with cooking spray and season with salt and pepper. Place squash, cut-side down, on the prepared baking sheet.
Bake in the preheated oven until tender, about 35 minutes. Cool for 10 minutes.
Heat olive oil in a large skillet over medium heat. Saute onion in hot oil until just starting to turn translucent, about 5 minutes. Add asparagus to onion and continue to saute until onion is tender, 5 minutes more. Stir garlic into onion mixture; saute until fragrant, about 1 minute. Remove onion mixture to a plate.
Return skillet to heat. Scrape spaghetti squash flesh from skin using a fork to create 'noodles'; stir noodles into hot skillet and saute until heated through, 2 to 3 minutes.
Pour vegetable broth into skillet with spaghetti squash; bring broth to a simmer and cook until mostly reduced, about 2 minutes. Stir onion mixture and goat cheese into spaghetti squash until cheese is melted, about 1 minute. Add basil and thyme. Season with salt and ground black pepper to taste.