In a small bowl, mix together the paprika, garlic powder, and 1 teaspoon salt.
In a separate medium bowl, whisk together the minced garlic, balsamic, and rosemary. Stream in the olive oil, whisking until well incorporated.
Add the potatoes and carrots to a 6-quart slow cooker. Add half the spice mixture (about 1 ½ teaspoon) and 1 tablespoon balsamic sauce, tossing to coat.
Sprinkle the remaining spice mixture on the chicken, rubbing to cover the bird and under the skin.
Place the chicken on top of the vegetables. Pour over 1 tablespoon balsamic sauce. Set aside the remaining balsamic sauce for serving.
Cover and cook until breast registers 165 degrees and thighs register 175 degrees, 3-4 hours on high or 5-6 hours on low.
Remove the chicken from the slow cooker. It will be so tender that it may fall apart, though using two spatulas helps the process. If you’d like the chicken to have crispy skin, place it on a baking sheet and broil on high for 3-4 minutes.
Serve the chicken along with the potatoes and carrots, sprinkled with rosemary, and with the extra balsamic sauce on the side for drizzling or dipping.
Notes
For a prettier presentation, place the chicken breast-side up. For juicer breast meat, place the chicken breast-side down.Strain the liquid left in the bottom of the slow cooker and use it to drizzle over the chicken in addition to the balsamic sauce, or as a base for soups or stews.If you're short on time or just want a basic option, skip the balsamic sauce and simply use the dried spice mix.Carefully transfer the chicken to a sheet pan and broil on high for a few minutes if you want the skin to be crispy.Serve the chicken whole, family-style, or shredded with two forks.Store or freeze leftover shredded chicken for salads later in the week.