Combine ingredients in slow cooker: Add tomatoes (with juices), diced onion, minced garlic, diced carrots, diced celery, tomato paste, broth, dried basil, thyme, bay leaf, sugar, salt, and pepper to a 6-quart slow cooker. Stir to combine.
Cook: Cover and cook on low for 6–8 hours or on high for 3–4 hours. Vegetables should be very soft.
Remove bay leaf: Discard the bay leaf.
Blend the soup: Use an immersion blender to puree the soup directly in the slow cooker until smooth. Alternatively, transfer in batches to a countertop blender and return to the pot. For ultra-smooth texture, strain through a fine-mesh sieve.
Finish with cream and basil: Stir in heavy cream, half-and-half, or coconut milk. Add fresh basil leaves. Taste and adjust seasoning with salt, pepper, and additional sugar if needed. If the soup is too acidic, add a pinch of baking soda.
Serve: Ladle into bowls and garnish with croutons, Parmesan, a drizzle of cream, or extra fresh basil.