For the sauce: In a large pot on medium high heat add the oil and onions. Season with a pinch of salt and a few grinds of pepper. Cook, while stirring, until tender, about 5 minutes. Add the paprika, garlic and whole sprigs of thyme. Stir until fragrant, about 5 minutes. Add the brown sugar, ketchup, vinegar, mustard and tomatoes and stir to combine. Add the cola and scrape the bottom of the pot with a wooden spoon to remove any bits that are stuck. Cook, stirring occasionally, until the sauce thickens to slightly thinner than the consistency of ketchup, about 45 minutes.
For ribs: Heat half of a grill to 425 degrees F. Sprinkle the ribs all over with salt and pepper. With the grill lid open, place the ribs over the direct heat side of the grill and grill to get char marks on all sides, turning and flipping the ribs as needed. Lower the heat to 225 to 250 degrees F and move the ribs to the indirect side of the grill and baste with the some of the sauce. Cover and cook, quickly basting and recovering every 30 minutes, until the meat is tender, 2 to 4 hours depending on the size of the ribs, or until the internal temperature is 200 degrees F.