Set the Instant Pot to saute mode and add the olive oil and butter. Once the butter melts, add the diced onion and cook for 4 to 5 minutes until soft and translucent.
Add the garlic and tomato paste. Stir for 1 minute until the paste darkens slightly and becomes fragrant. Pour in the vegetable broth and stir, scraping up any bits from the bottom.
Add the whole peeled tomatoes with their juices, diced tomatoes with their juices, sugar, salt, and black pepper. Stir to combine. Press cancel.
Lock the lid, set the valve to sealing, and cook on high pressure for 10 minutes. When the cook time is done, let the pressure release naturally for 10 minutes, then carefully turn the valve to venting.
Add the fresh basil leaves to the pot. Using an immersion blender, blend the soup directly in the pot until completely smooth and silky. Alternatively, carefully transfer the soup in batches to a countertop blender and blend until smooth.
Stir in the heavy cream until fully incorporated. Taste and adjust salt and pepper as needed. Ladle into bowls and garnish with extra basil leaves. Serve with grilled cheese on the side.