Preheat a griddle or 2 large skillets over medium heat.
In a small bowl, stir together the softened butter, salt, and smoked paprika.
Place the bread slices on a cutting board. Spread each slice with ½ tablespoon of the smoked paprika butter.
Place four slices of bread, butter side down, on the griddle.
Top each with a slice of provolone, then ¼ of the pulled pork.
Top the pulled pork with ¼ of the cheddar and ¼ of the gouda on each sandwich.
Place the remaining slices of bread on top, butter side up.
When the bread is golden brown on the bottom, carefully flip each sandwich and continue cooking until the bread is golden brown on the bottom and the cheese is melted, about 2–3 minutes.