Rinse chicken and pat dry with paper towels. Stir brown sugar with salt, paprika, garlic powder and cayenne in a small bowl. Rub spice mixture all over chicken, pressing gently so spices stick.
Heat oil in a 6- to 7-quart pressure cooker over medium. When oil is very hot, add chicken, breast-side down, using tongs. Cook until golden brown all over, 1 1/2 to 2 min per side. Remove chicken to a cutting board. Discard hot oil from pot. Lay a rack in the bottom of the pot. Pour in water, then arrange potatoes on the rack. Lay chicken, breast-side up, over potatoes. Cover, and lock pressure cooker lid according to manufacturer’s instructions. Increase heat to high. When steam starts to come out of the vent pipe, reduce heat to medium. Cook for 20 min. Remove from heat and let pressure drop naturally before unlocking lid.
Transfer chicken to a platter. Toss potatoes with butter and dill in a bowl. Transfer to platter. Remove rack from pot. Return pot with leftover juices to stovetop over medium-high. Bring to a boil and reduce sauce to 1 cup, 10 to 15 min. Season with pepper. Serve with chicken and potatoes.