In a medium bowl, combine the shredded chicken with BBQ sauce, salt, and pepper. Mix thoroughly until each piece is well-coated.

Place a medium skillet over medium heat and add 1 tablespoon of butter, allowing it to melt.

Add one slice of sourdough bread to the skillet, layer half of the mozzarella cheese, spread the BBQ chicken mixture, then add the remaining mozzarella and cheddar cheese, followed by the second slice of bread.

Cook the sandwich for about 3-4 minutes until the bottom is golden brown and cheese starts to melt.

Carefully flip the sandwich, add another tablespoon of butter, and cook the other side for another 3-4 minutes until golden and crispy.

Allow the sandwich to cool for a minute before slicing in half and serve immediately.