Gather and prepare all ingredients: dice onion, carrot, celery, mince garlic, and crush canned tomatoes.

Melt 2 tablespoons butter with 2 tablespoons olive oil in a Dutch oven over medium heat. Cook diced vegetables for 8-10 minutes.

Add minced garlic, crushed tomatoes, and vegetable broth to the pot, mixing everything thoroughly.

Allow the mixture to come to a gentle simmer, then reduce heat and cook for 25-30 minutes.

Blend the soup with an immersion blender until smooth, adjusting texture with more broth if needed.

Stir in heavy cream and chopped basil, warming through gently.

Prepare grilled cheese by mixing cheeses and buttering bread slices, grilling until golden brown.

Serve soup hot with grilled cheese on the side, cut into strips for dipping.