If grilling outdoors, clean the grill grates and preheat to medium heat. If grilling indoors, use a
sauté pan or indoor grill preheated to medium heat. Coat grill in oil.
Place quartered tomatoes in a large bowl. Add two tablespoons of olive oil, salt and pepper. Toss to coat.
Place the tomatoes on the grill, or in a grill tray, until they char (if using outdoor grill) or caramelize (indoor grill/sauté pan), approximately five minutes on each side.
Allow tomatoes to cool slightly, then place in food processor or use immersion blender to puree.
In a large sauce pot on medium heat, heat remaining 2 tablespoons of olive oil, garlic and shallot. Allow to cook for approximately three minutes or until garlic and shallot caramelize.
Add tomato puree and basil, and allow them to cook with garlic, shallot and olive oil for approximately three minutes.
Add chicken broth, and reduce to low heat. Allow ingredients to simmer uncovered on low heat for at least 30 minutes.
While soup cooks, cut French bread loaf into one inch cubes and toss with 2 tablespoons of olive oil and salt and pepper to taste. Place cubes on a sheet pan and place in the oven until toasted.
Using a ladle, portion soup into bowls and top with fresh croutons and Cheddar Jack cheese.