Prepare a charcoal grill for indirect grilling or heat a gas grill to medium heat. Brush drumsticks with oil and season with salt and pepper. Place on the grill over the coals (if using charcoal grill) and grill until golden brown on all sides. Move the drumsticks to the cooler part of the grill, place the lid on and continue cooking until just cooked through, about 45 minutes to 1 hour, brushing with the glaze during the last 20 minutes of cooking. Remove from the grill, brush with more of the glaze and let rest 10 minutes before serving. If using gas, cook at medium heat the whole time with the lid closed after a crust has formed.
Heat oil in a medium saucepan over medium-high heat on the grates of the grill. Add the shallots and garlic and cook until a light golden brown. Add vinegar and cook until reduced by half. Add sugar, paprika, tomatoes and chicken stock and cook until reduced by half and tomatoes are very soft. Transfer to a blender and blend until smooth. Return to the saucepan and cook until thickened; season with salt and pepper. Let cool.