1. Prepare Herb Mayonnaise: Combine all ingredients. Chill until ready to use.
2. Prepare Sandwiches: Spread Herb Mayonnaise on 1 side of each bread slice. Layer 4 bread slices, mayonnaise sides up, each with 1 cheese slice, 1 Tbsp. red bell peppers, 1 rounded tablespoonful artichoke hearts, and 1/4 cup arugula. Top with remaining cheese and bread slices, mayonnaise sides down. Spread butter on outside of each sandwich.
3. Cook sandwiches in a large nonstick skillet or griddle over medium-high heat 3 to 4 minutes on each side or until lightly browned and cheese is melted. Serve immediately.