Prepare the slow cooker by drizzling about 1 tablespoon of oil into the bottom. Lay the chicken on top of the oil. In a medium bowl, whisk together the remaining oil, balsamic vinegar, tomato paste, brown sugar, dried, basil, garlic powder, and salt.
Pour over the chicken. Cover the lid on the slow cooker and cook on low for 6 hours or until the chicken is fork tender and reaches 165 F. in the thickest part.
Remove the lid and stir the sauce. Place a mound of cheese over each chicken breast. Re-cover until the cheese has melted.
Prior to serving, you can lay one slice of tomato over each chicken breast and sprinkle with fresh or dried basil for a caprese style meal.