Heat the olive oil in a large skillet over medium heat. Add the sliced red onion and cook until softened, about 2 minutes. Stir in the minced 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 to 3 minutes. Remove from heat and set aside.

In a small microwave-safe bowl, combine the butter, minced garlic, and chopped parsley. Microwave until the butter is melted, about 30 seconds.

Lay the 12 slices of sourdough bread on a work surface. Brush the melted garlic butter mixture over one side of each slice.

Flip half the slices so the buttered side is down. Top each of those 6 slices with a slice of Havarti cheese, then divide the BBQ chicken mixture evenly among them. Place another slice of Havarti on top of the chicken, then cover with the remaining 6 slices of bread, buttered side up.

Heat a large non-stick skillet over medium heat. Place the sandwiches in the skillet (you may need to work in batches) and cook until the bottoms are golden brown, about 3 to 5 minutes. Flip the sandwiches, reduce the heat to medium-low, and continue cooking until the other side is browned and the cheese is fully melted, another 3 to 5 minutes. Serve hot.