Build the onion trivet. Scatter the thick onion rounds over the bottom of a 6-qt slow cooker in an even layer. Add the potatoes and carrots around the edges if using; toss them with the olive oil, salt, and pepper first.

Season the chicken. Remove giblets and pat the chicken very dry with paper towels (dry skin = better browning later). Stuff the cavity with the halved lemon, rosemary, and thyme. Rub the softened butter all over the outside.

Mix and apply the rub. Stir the paprika, garlic powder, onion powder, dried thyme, salt, pepper, and brown sugar in a small bowl. Sprinkle evenly over the entire chicken, patting to adhere.

Slow cook. Set the chicken breast-side up on the onion trivet. Cover and cook on LOW 6 to 6.5 hours (or HIGH 3 to 3.5 hours), until the thigh reads 165°F (74°C) on an instant-read thermometer. Do not add water — the onions release plenty of juices.

Crisp the skin. Heat the broiler to high with a rack 6 inches from the element. Transfer the chicken to a foil-lined baking sheet, brush with a spoonful of the slow-cooker juices, and broil 4 to 6 minutes until deep golden and crisp, watching constantly.

Rest and serve. Rest 10 minutes before carving so the juices redistribute. Serve with the softened onions and vegetables from the cooker, spooning over the cooking juices.