Season the chicken breasts with salt and pepper on both sides. In a skillet over medium-high heat, warm the olive oil. Add the chicken breasts and cook for about 6-7 minutes on each side until fully cooked through. Let them rest for a few minutes before slicing thinly.
While the chicken rests, slice the avocado and prepare the cheese slices.
Spread a thin layer of mayonnaise and Dijon mustard on each slice of bread.
Layer sliced chicken, avocado, and cheese on two slices of bread. Add thin slices of red onion and tomato, then top with the remaining slices of bread, mayo side down.
Heat a skillet or grill pan over medium heat. Add butter or olive oil, then place the assembled sandwiches in the skillet. Cook until both sides are golden brown and the cheese is melted, about 3-4 minutes per side.
Remove the sandwiches from the skillet and allow to cool for about a minute before slicing in half.