Combine boiling water and tomatoes in a bowl; cover and let stand 30 minutes or until soft. Drain and finely chop.
Heat 1 teaspoon oil in a large nonstick skillet over medium heat. Add 1/3 cup shallots, sugar, and garlic; cook 4 minutes or until lightly browned, stirring frequently. Spoon into a bowl; stir in 1 1/2 teaspoons vinegar.
Combine chopped tomatoes, shallot mixture, cheese, basil, and 1/4 teaspoon salt, stirring well.
Cut a horizontal slit through thickest portion of each chicken breast half to form a pocket. Stuff about 2 tablespoons cheese mixture into each pocket. Sprinkle chicken evenly with 1/2 teaspoon salt and black pepper.
Heat 1 teaspoon oil in pan over medium-high heat. Add chicken; cook 6 minutes on each side or until done. Remove chicken from pan; cover and keep warm. Add broth, remaining shallots, 2 tablespoons vinegar, and thyme; bring to a boil. Combine cornstarch and water, stirring with a whisk. Add cornstarch mixture to pan; bring to a boil. Cook 1 minute or until sauce is slightly thick, stirring constantly. Serve sauce over chicken.
Stuffed chicken breasts are a great way to add flavor and retain moisture in chicken. Use a paring knife to cut the small pocket into the chicken breasts.