Place a rack in the center of the oven and preheat to 325°F (163°C).
Pat the chicken quarters very dry with paper towels and arrange them on a rimmed baking sheet or roasting pan with space so pieces do not overlap (some touching at the edges is fine).
In a bowl or liquid measuring cup, whisk together the olive oil, kosher salt, paprika, smoked paprika, ground cumin, garlic powder, cayenne, and black pepper until combined.
Rub the spice mixture all over each chicken quarter, coating both the skin and underside. Arrange the pieces skin-side up on the pan.
Roast in the preheated oven until a kitchen thermometer inserted into the thickest part of the thigh (avoiding bone) reads at least 165°F (74°C), about 1 to 1½ hours; dark meat can be cooked longer if desired.
Transfer the chicken to a carving board, tent loosely with foil, and let rest for 10 minutes.
Serve the chicken with pan juices spooned over the top or use bread to mop up the juices, if desired.