Spread a thin layer of mayonnaise on one side of each bread slice.
On two of the slices, layer the sliced chicken breast followed by the tomato slices.
Season the tomatoes with salt and pepper, then top with avocado slices.
Place two slices of Monterey Jack cheese over the avocado on each sandwich and cover with the remaining bread slices, mayonnaise-side down.
Spread the exterior sides of the sandwich with the softened butter.
Preheat a non-stick skillet or griddle over medium heat.
Place the sandwiches in the skillet and cook for 3 to 4 minutes per side, or until the bread is golden brown and the cheese has fully melted.
Remove from heat, slice diagonally, and serve immediately.