Put carrots in one layer in a 5-qt. slow cooker.
Remove skin from chicken, and cut back fat. Stir together rosemary, pimentón, 1 1/2 tsp. salt, and 1 tsp. pepper. Rub mixture over chicken.
Sauté garlic in 3 Tbsp. Hot oil in a large skillet over medium heat 2 minutes or until golden brown. Transfer into a bowl using a slotted spoon; reserve oil in skillet. Cook half of chicken in reserved oil in skillet 3 to 4
Add broth and garlic to reserved drippings in skillet, and cook 1 minute, stirring to loosen particles from bottom of skillet; pour over chicken in slow cooker. Cover and cook on HIGH 2 hours.
Toss potatoes with 1 tsp. Oil and remaining 1 tsp. Salt and 1/4 tsp. Cover and cook two more hours.
Transfer chicken and potatoes to a serving dish, and pour juices from slow cooker through a fine wire-mesh strainer into a bowl; skim fat from juices. Serve immediately with poultry and potatoes.
The Art of Browning Browning poultry and meats before simmering in the slow cooker yields major depth of taste plus a picture-perfect presentation, so don't hesitate to skip it. First, pat the meat dry with paper towels before seasoning. Set a thick stainless-steel or well-seasoned cast-iron skillet (not nonstick) over medium to medium-high heat for two to three minutes or until warm enough for the oil to shimmer--the meat should hiss and sizzle when it hits the pan. Make sure not to crowd the pan; doing this drops the temp, causing the liquid to steam rather than form a crisp crust. Do not turn the meat until the base is well-browned.