1. Preheat the oven to 350° F degrees with the blower on set on low.
2. Drain the tomatoes through a strainer, reserving the juice.
3. Place drained tomatoes in a single layer on a sheet and place in the preheated oven. Roast in the oven for 25 minutes to remove moisture and intensify flavor.
4. Place the olive oil in a large saucepan over low heat and add the onions, fennel and carrots with 2 tsps. of the kosher salt. Sweat under cover for 15 minutes, stirring occasionally until veggies are soft and have released their juices.
5. Remove cover, add garlic, and cook uncovered 5 additional minutes over medium heat, stirring occasionally to prevent sticking.
6. Add the white wine and stir, deglazing flavors from bottom of pan.
7. Add the roasted tomatoes and reserved tomato juice to vegetables and bring to a boil. Reduce heat to a simmer, and cook for 30 minutes.
8. Remove the soup from the heat and stir in the basil leaves. Allow soup to cool for 30 minutes.
9. Puree soup in small batches using a blender or food processor.
10. Correct seasoning with salt and pepper.
11. Place the soup in a saucepan and bring to a boil. Divide the soup evenly into 12 hot soup bowls and garnish with the grilled cheese sandwiches.
You may complete this all way till step 10, at least one day in advance. Chill the soup rapidly and then complete step number 11 immediately prior to serving.