Heat panini maker or stovetop panini grill according to manufacturer’s directions.
Brush one side of bread slices with oil. Place bread, oil side down, on work surface.
In a small bowl, mix Italian dressing and mayonnaise. Spread on bread slices on work surface.
Top each bread slice with one slice cheese. Top one side of each sandwich with 2 slices ham and 15 slices pepperoni. Top the other side of each sandwich with 2 slices tomato, half the roasted red bell pepper strips, and half of the pepperoncini.
Put the 2 sandwiches together.
Grill sandwiches in panini maker 5-7 minutes or until bread is browned and cheese is melted.