Spread 1/2 Tbsp butter over one side of each piece of bread. Spread 1 Tbsp pesto over the opposite side of each piece of bread. The butter sides will be facing out on the sandwich and the pesto sides facing in.
Cut the sun dried tomatoes into small pieces. Sprinkle them over one of the pesto coated surface of bread. Add the mozzarella on top of the sun dried tomatoes (shred or break the mozzarella into small pieces to facilitate fast melting). Place the other piece of bread on top, pesto side facing in.
Place the sandwich in a small skillet and heat over medium-low until golden brown on one side. Flip and cook until golden brown on the other side, and the cheese is melted on the inside (about 3-5 minutes each side). Serve hot.