Spread 2 tablespoons of Nutella each evenly over 4 slices of bread, then top with 2 slices of stracchino. Make 4 sandwiches with the remaining slices of bread.
Prepare a charcoal or gas grill for medium-hot cooking or place a grill pan over medium-high heat.
Brush both sides of each sandwich with melted butter.
Grill the sandwiches until light golden brown on both sides, 2 to 3 minutes on each side.