For aioli, place the first 8 ingredients in a blender; cover and process until mixture is smooth.
Brush the vegetables with oil. Place in broiling pan in oven and broil 3-4 inches from heat, or grill, covered, over medium heat until tender, 4-5 minutes per side. Chop the peppers when cool enough to handle.
Spread cut sides of each roll with 1 tablespoons aioli. Layer bottoms with 1 slice cheese, 2 slices eggplant, peppers and remaining cheese. Replace tops.
In a panini maker, grill sandwiches until the cheese is melted, 5-7 minutes. Serve remaining aioli with sandwiches or save for another use.