In large, skillet over medium heat, warm 3 Tbs. olive oil. Add onions, cover and cook, stirring occasionally, until they are completely limp, 10 to 15 minutes. Uncover pan, reduce heat to medium-low and cook, stirring frequently, until onions are golden brown and sweet.
Preheat panini maker or griddle. Brush one side of each bread slice with olive oil. Lay slices, oiled side down, on clean work surface. Spread top of each slice with 1 TBS mayonnaise, add sliced chicken, and season with salt and pepper. Top with 1/4 cup caramelized onions and cheese. Top with other bread slice, oiled side up. Repeat with remaining ingredients.