Smoked gouda, while having a wonderful and inimitable flavor, has trouble achieving that gooey, iconic cheesy stretch that makes grilled cheese sandwiches so pleasing. By adding a partner cheese with a less assertive flavor and higher melting property, like deli muenster, you can achieve both melting and flavor perfection.
Lay out 4 slices of the bread on a work surface. Layer each with some of the smoked gouda, 1 slice of the muenster, a thin layer of sliced red pepper (about 2 tablespoons), 1/2 cup of the baby arugula and another layer of smoked gouda. Close the sandwiches with the remaining bread. Thinly and evenly spread the butter on both sides of the bread using about 1 1/2 tablespoons of the butter for each sandwich.
Place 2 sandwiches into a medium nonstick skillet over medium heat. Cover the skillet with a lid and cook until the bread is evenly golden brown, about 2 minutes. Flip, cover again and cook until the bread is golden brown and the cheese is visibly melted, about 2 minutes more. Repeat with the remaining sandwiches.