Heat the olive oil in a large skillet over medium heat. Cook the sliced red onion until soft, about 2 minutes, then add the minced garlic and cook until fragrant, about 30 seconds. Stir in the shredded chicken, paprika, Worcestershire sauce, and BBQ sauce, and cook until everything is heated through, 2-3 minutes. Set aside.

In a medium microwave-safe bowl, combine the butter, minced garlic, and parsley. Microwave until melted, about 30 seconds, stirring halfway. Lay out 12 slices of bread on a work surface and brush each slice with the melted garlic butter on one side only.

Flip the bread slices over so the buttered side is down. On 6 slices, layer one slice of Havarti cheese, a portion of the BBQ chicken mixture, and then another slice of Havarti cheese. Top with the remaining 6 bread slices, buttered side up.

Place the sandwiches in a large non-stick skillet over medium heat. Cook until the bottoms are golden brown, 3 to 5 minutes. Carefully flip the sandwiches, reduce the heat to medium-low, and continue cooking until the second side is browned and the cheese is fully melted, 3 to 5 minutes longer. Serve immediately.