Preheat oven to 375 degrees. Season cut sides of squash with 1/4 teaspoon salt. Bake, cut sides down, on a baking sheet until tender, about 45 minutes. Let stand until cool enough to handle. Scrape flesh into strands using a fork. Let drain, covered to keep warm, in a colander set over a bowl.
Meanwhile, heat 1 teaspoon oil in a medium skillet over medium heat. Cook onion and garlic, stirring, until soft but not brown, 8 to 10 minutes. Add oregano; cook until fragrant, about 30 seconds. Divide mixture between 2 bowls.
Stir 1 tablespoon plus 1 1/2 teaspoons cheese into 1 bowl of onion-garlic mixture. Mix in breadcrumbs, turkey, 1/2 teaspoon salt, and 1/4 teaspoon pepper using your hands. Form mixture into 12 balls.
Heat 2 teaspoons oil in a large skillet over medium heat. Brown meatballs, turning, 4 to 6 minutes; transfer to a plate. Add remaining teaspoon oil to skillet. Cook mushrooms, 1/4 teaspoon salt, and 1/8 teaspoon pepper, adding a few tablespoons water if skillet browns too quickly, until soft and beginning to brown, 6 to 8 minutes. Add remaining bowl of onion-garlic mixture, meatballs, and stock; bring to gentle simmer. Cook, partially covered, until meatballs are cooked through, 4 to 5 minutes. Add spinach; cook until just wilted, about 1 minute.
Divide squash and meatball mixture into 4 servings. Sprinkle with remaining cheese.