1. Preheat oven to 450°F. Set 2 fine-mesh strainers over 2 bowls. Line a large baking sheet with parchment. Divide dough into 4 portions; roll them into 7-inch rounds.
2. Warm 1 Tbsp. oil in a skillet over medium heat. Cook sausage, stirring and breaking up meat, until browned, 5 to 10 minutes. Transfer to a strainer to drain.
3. Warm 1 Tbsp. oil in same skillet over medium heat. Cook spinach, stirring, until wilted, about 3 minutes. Season with salt and pepper. Remove from heat. Transfer to second strainer; squeeze out liquid.
4. Spread pumpkin on one side of each dough round, leaving a 1/2-inch border. Divide sausage, spinach and cheese among rounds, scattering over pumpkin. Fold dough over filling and pinch edges to seal.
5. Brush tops of calzones with 1 Tbsp. oil. Transfer to baking sheet; bake until golden, about 15 minutes. Allow to cool slightly before serving.