Arrange the chicken breasts in a single layer on the bottom of the slow cooker. Pour ½ cup of chicken broth into the slow cooker. Cover and cook on low for 3-4 hours.
Remove the chicken from the slow cooker and shred using two forks. Drain the chicken broth from the slow cooker.
Return the chicken to the slow cooker and stir in the Caesar salad dressing, parmesan cheese, parsley, and ground black pepper. Cook on low for 30 minutes.
Heat a skillet or griddle to medium-low heat. Spread butter onto the cut side of each bun. Place cut-side down into the pan and cook until the buns are toasted.
Serve the shredded chicken on the buns with romaine lettuce.