Turn on the grill to preheat to medium heat. In a large bowl, combine chopped cheese, softened butter, chopped parsley, chopped garlic, garlic powder, and paprika. Beat everything with a mixer until fluffy, homogeneous.
Grease each baguette slice liberally with the oil mixture on all sides. Re-assemble the baguette from the slices and wrap it in a dense foil layer, pressing the foil tightly against the baguette.
Place the baguette on the grill rack, cover the grill, and cook for about 15-20 minutes, turning the baguette once during the entire cooking time.