Spritz 2 pieces of newspaper lightly with vegetable oil and place in the bottom of a charcoal chimney starter. Fill the chimney starter with natural chunk charcoal, 2 to 3 pounds, and set on the charcoal grate of a kettle grill until hot and ashy, approximately 15 to 20 minutes.
Meanwhile, butter the bread slices on each side. Combine the cheeses, mustard, paprika and pepper in small bowl.
Fold a long (24-inches) piece of heavy-duty aluminum foil in half. Set a large metal griddle spatula in the center and fold the sides up around the spatula, forming a tray. Spritz the spatula-tray with vegetable oil. Repeat with a second piece of aluminum foil and another griddle spatula. Divide the cheese mixture evenly between the spatula-trays, and set aside. Set aside two additional 15-inch sheets of heavy-duty aluminum foil.
Carefully and distribute the hot charcoal onto one side of the charcoal grate. Set the cooking grate in place and heat for 2 to 3 minutes.
Set the cheese-filled spatula-trays on the grill over indirect heat. Cook for 6 to 9 minutes or until the cheese melts and bubbles around the edges. You may have to adjust the placement of the spatula-trays to ensure even melting and keep the cheese from overheating and breaking.
Grill the bread for 1 to 2 minutes per side over direct heat.
Place 1 slice of bread on each of the reserved sheets of aluminum foil. Pour off excess grease into a bowl, if desired, then drizzle one spatula‚Äôs worth of cheese onto one slice of bread and top with the a second slice of bread. Fold the foil around the sandwich. Repeat with remaining cheese and bread slices, and then return the sandwich packets to the grill over indirect heat for 1 to 2 minutes.
Unwrap and serve immediately