In a slow cooker, combine the crushed tomatoes, chopped onion, minced garlic, vegetable broth, dried oregano, sugar, salt, and pepper.
Stir in the olive oil to enhance the flavors.
Cover and cook on low for 6-8 hours or on high for 3-4 hours until the vegetables are tender and the flavors meld together.
Once the soup is cooked, blend it with an immersion blender directly in the slow cooker until smooth. If you prefer a chunkier texture, blend only half.
Stir in the heavy cream or coconut milk and the chopped basil leaves. Mix well and let it sit for an additional 15-20 minutes on low heat to combine the flavors.
Taste and adjust the seasoning if necessary.
Serve hot, drizzled with a bit of olive oil or topped with grated Parmesan cheese if desired.
Notes
For a dairy-free option, use coconut milk instead of heavy cream.