Preheat oven to 350 degrees F (175 degrees C). Place spaghetti squash cut-side down in a nonstick baking pan.
Bake spaghetti squash in the preheated oven until tender when pierced with a fork, about 45 minutes. Cool in the pan for 5 minutes. Transfer to a cooling rack; cool until easily handled.
Heat 1 tablespoon olive oil in a large skillet over medium heat. Add onion and garlic; cook and stir until onion softens, 3 to 5 minutes. Stir in chicken breasts, tomatoes with juice, Italian seasoning, and paprika; cook until chicken is no longer pink in the center, 15 to 20 minutes. Remove from heat; spoon out excess liquid.
Scrape squash flesh into spaghetti strands using a fork. Stir squash strands into the skillet. Fold in remaining 1 tablespoon olive oil with a spatula. Season with salt and pepper. Serve in a pasta bowl garnished with Parmesan cheese and oregano.