BBQ Chicken: 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, paprika, Worcestershire sauce, and BBQ sauce. Stir to combine and cook until heated through, about 2-3 minutes.
Garlic Butter Bread: Melt butter, garlic, and parsley in a medium bowl in the microwave. Lay 12 slices of bread on a work surface. Spread the melted butter mixture over each piece. Flip each slice over. On 6 slices, layer a cheese slice, an even portion of the BBQ chicken mixture, and another slice of cheese. Top with the remaining bread slices, buttered side out.
Cook the sandwiches in a large non-stick skillet over medium heat until the bottoms are browned, about 3-5 minutes. Flip and continue cooking on medium-low until the other side is browned and the cheese is melted, another 3-5 minutes.