Make the barbecue sauce: Bring the cola to a boil in a medium saucepan over high heat. Cook until reduced by about half, about 5 minutes. Add the ketchup, vinegar, brown sugar, mustard, garlic, Worcestershire sauce, cayenne and 2 teaspoons paprika. Bring to a simmer and cook, stirring occasionally, until thickened, 8 to 10 minutes. Remove from the heat and let cool slightly.
Preheat a grill to medium and prepare for indirect grilling: On a charcoal grill, bank the coals to one side; on a gas grill, turn off half the burners. Combine 2 tablespoons salt, 1 tablespoon black pepper and the remaining 1 tablespoon plus 1 teaspoon paprika in a bowl. Rub the salt mixture all over the ribs. Remove 1 cup of the barbecue sauce to a small bowl; brush over the ribs.
Wrap each rack of ribs separately in foil. Put the racks, overlapping if necessary, on the cooler side of the grill (indirect heat). Cover and cook, flipping the ribs after 30 minutes, until just tender and beginning to pull away from the bones, about 1 hour to 1 hour 15 minutes.
Remove the ribs from the grill and carefully unwrap; pour any juices into the pan with the remaining barbecue sauce. Bring the sauce to a simmer over medium heat and cook until thickened, 5 to 7 minutes.
Increase the grill heat to medium high. Brush the ribs with some sauce from the pan and grill over direct heat, flipping every 5 minutes, until glossy and crisp at the edges, about 20 minutes. Transfer the ribs to a large cutting board and let rest 5 minutes. Cut the ribs between the bones and serve with the remaining barbecue sauce.
If you're making the ribs on a charcoal grill, you'll need 3 to 4 pounds of charcoal. Add several new briquettes to the coals every 30 minutes to maintain the heat.