Heat the olive oil in a large pot over medium heat. Add the diced onion and cook until soft, about 5 minutes.
Stir in the minced garlic and cook for 1 minute until fragrant.
Pour in the crushed tomatoes and vegetable broth. Bring the mixture to a simmer and cook for 10 minutes.
Remove the pot from the heat. Use an immersion blender to puree the soup until smooth.
Return the pot to low heat. Stir in the heavy cream and chopped basil. Season with salt and black pepper.
While the soup simmers, spread softened butter on one side of each bread slice. Place cheese between two slices, buttered sides out.
Cook the sandwiches in a skillet over medium heat until golden on both sides and the cheese melts.
Cut each grilled cheese into strips to create dippers and serve them alongside the soup.