Spray a 6-quart slow cooker with non-stick cooking spray. Place sweet potatoes in slow cooker in an even layer, pour over chicken broth then sprinkle with half of the garlic and season with salt and pepper.
In a small mixing bowl whisk together paprika, brown sugar, onion powder, thyme, rosemary and cayenne pepper. Sprinkle half of the spice mixture evenly over potatoes, set remaining aside.
Heat olive oil in a large pot over medium-high heat. Dab skin side of 3 pieces of the chicken dry with paper towels and season both sides with salt and pepper. Sear in oil, skin side down, until golden, about 4 minutes. Transfer to slow cooker over sweet potato layer. Repeat with remaining 3 pieces of chicken.
Sprinkle remaining spice mixture and garlic evenly over chicken, the cover slow cooker and cook on low heat until sweet potatoes are tender and center of chicken registers 165 on an instant read thermometer, about 4 1/2 - 5 hours.
Add broccoli either during the last 30 minutes of cooking or steam in bag as directed on package, season with salt. Plate chicken, sweet potatoes and broccoli and serve with a few spoonfuls of the sauce from the slow cooker (which you can skim off a little of the fat first if preferred).