Preheat a griddle over low heat.
Spread a thin layer of mayonnaise on one side of each slice of bread¿this will give the bread an even golden color when griddled. Flip 4 slices over and layer each with 1 slice Monterey Jack, 1 slice tomato, 1 slice red onion, 1 slice avocado and 1 more slice Monterey Jack. Top with another slice of bread, mayonnaise side up.
Place the sandwiches on the preheated griddle and toast until the bread is golden brown and the cheese is melted, about 2 minutes per side.