Preheat the panini press to 350° or medium-high, depending on the type of grill you have.
In a small dish, mix the mayonnaise, mustard, and lemon juice together and set aside.
Slice the chicken breasts as thin as possible, slice the cheese and onion. Place the peppers on a small plate lined with paper towels and pat the liquid off of them.
Spread each slice of bread with the mayo mixture and arrange the chicken, cheese, onion, and peppers on four of the slices of bread. Add the second bread slice on top and place in the heated panini press.
Cook for 5 minutes or until the cheese melts and the bread is golden and crisp. Transfer to a cutting board, cut in half, and serve.