Place chicken breasts in a shallow dish. Coat with Italian dressing. Cover with plastic wrap and let marinate in the refrigerator, 8 hours to overnight.
Heat a grill pan over medium-high heat. Remove chicken breasts from marinade and place in the skillet; cook until no longer pink in the center and the juices run clear, about 6 minutes per side.
Slice chicken breasts; lay Brie cheese on top to melt.
Preheat oven to 300 degrees F (150 degrees C).
Heat 1 teaspoon olive oil in an oven-safe skillet over medium heat. Add tomatoes and 1/8 teaspoon salt; cook, stirring once, until tomatoes release their juices, about 4 minutes. Remove from heat and stir in 2 tablespoons balsamic vinegar and basil.
Transfer skillet to the preheated oven; bake tomatoes until softened, about 15 minutes.
Combine remaining 2 teaspoons olive oil, 1 tablespoon balsamic vinegar, and 1/8 teaspoon salt in a bowl. Add spinach; toss to coat.
Mix mayonnaise, Dijon mustard, garlic, and pepper together in a bowl.
Place bottom half of the loaf on a baking sheet. Spread mayonnaise mixture on top. Cover with spinach and tomatoes. Place chicken with Brie cheese on top. Cover with top half of the loaf.
Bake in the preheated oven until heated through, 5 to 10 minutes. Cut into 4 sections.