Heat oil in a large skillet over medium heat. Cook onions until soft, about 2 minutes. Add garlic and cook until fragrant, about 30 seconds. Add the shredded chicken, smoked paprika, Worcestershire sauce, and BBQ sauce. Stir to combine and cook until everything is heated through, about 2-3 minutes.

Add butter, garlic, and minced parsley in a medium bowl. Microwave until melted.

Lay 12 slices of bread on work surface. Spread melted butter mixture over each piece.

Turn each piece over and 6 of the bread slices with 6 of the cheese slices and the BBQ chicken mixture, dividing evenly. Add the remaining cheese over the top of the chicken and top with the remaining 6 slices of bread (both the bottom and the top should have the buttered side exposed).

Set buttered sandwiches in a large non-stick skillet. Place over medium heat and cook until the bottoms of the sandwiches are browned, 3 to 5 minutes. Turn sandwiches over, reduce heat to medium-low and continue cooking until the bottom is browned and the cheese is melted, 3 to 5 minutes longer.