Heat olive oil in a large skillet over medium heat. Add thinly sliced red onion and cook until softened, about 2 minutes. Add minced garlic and cook until fragrant, about 30 seconds.
Add the shredded chicken, smoked paprika, Worcestershire sauce, and BBQ sauce to the skillet. Stir to combine and cook until everything is heated through, about 2–3 minutes. Remove from heat and set aside.
In a small bowl, combine butter, minced garlic, and minced parsley. Microwave until melted, about 30 seconds. Stir to combine.
Lay all 12 slices of bread on a clean work surface. Brush the garlic butter mixture evenly over one side of each slice. Flip each slice over so the buttered side faces down.
On 6 of the bread slices (buttered side down), place one slice of Havarti cheese. Top each with an equal portion of the BBQ chicken mixture. Add a second slice of Havarti on top of the chicken. Cap with the remaining 6 bread slices, buttered side facing up.
Place sandwiches in a large non-stick skillet over medium heat. Cook until the bottom is golden brown, 3–5 minutes. Flip each sandwich, reduce heat to medium-low, and cook until the second side is golden brown and the cheese is fully melted, another 3–5 minutes.
Remove from skillet, let rest 1–2 minutes, then slice in half and serve immediately.
