For the buffalo mayo and chicken: Mix together the mayonnaise, wings sauce and lemon juice in a bowl. Mix in the shredded chicken.
For the panini: Preheat a panini press or griddle.
Place a piece of cheese on each slice of bread. Place some of the shredded chicken, bacon and red onion rings on half of the slices. Cover with a top piece of bread. Brush oil on both sides of each sandwich. Cook the sandwiches in a panini press (or on a griddle with a foiled brick for pressing) until toasted on both sides.