Preheat oven to 350°F (177ºC). Boil 1 cup water in a small pan. Add sun-dried tomatoes, remove heat, and cover. Let stand for 30 minutes or until soft. Drain and finely chop.
Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Add shallots, sugar, and garlic. Cook until lightly browned, stirring often. Spoon shallots into a small bowl and combine with ½ tablespoon balsamic vinegar, chopped tomatoes, shallots, goat cheese, basil, and ¼ teaspoon salt.
Cut a slit into the side of the chicken breast to form a pouch for the stuffing. Stuff about 2 tablespoons cheese mixture into each pocket. Sprinkle chicken evenly with ½ teaspoon salt and black pepper.
Heat 1 tablespoon oil in the same large skillet used previously over medium heat. Add chicken and cook for 6 minutes on each side until brown. Remove from the pan and cook on a baking sheet in the oven for 10 minutes, five on each side, or until the chicken reaches a minimum internal temperature of 165ºF (73.9°C).
In the same pan the chicken was cooked in, add broth, 2 tablespoons balsamic vinegar, and thyme and bring to a boil. Dissolve cornstarch in the water, stirring with a fork. Add cornstarch mixture to the pan and cook one minute or until the sauce thickens, stirring constantly. Serve sauce over chicken.