In a food processor or blender, combine mayonnaise, red pepper, and garlic. Blend or process until mixture is almost smooth.
Preheat an electric sandwich press, a covered indoor grill, a grill pan, or a skillet. To assemble sandwiches, split each roll in half horizontally; spread cut sides of rolls with the mayonnaise mixture. Layer chicken, prosciutto, cheese, and mesclun on bottom halves of rolls. Replace top halves of rolls.
Place sandwiches (half at a time, if necessary) in the sandwich press or indoor grill; cover and cook about 6 minutes or until cheese melts and rolls are toasted. (If using a grill pan or skillet, place sandwiches on grill pan. Weight sandwiches down and grill about 2 minutes or until rolls are lightly toasted. Turn sandwiches over, weight down, and grill until remaining side is lightly toasted.) Makes 4 sandwiches.