Cook bacon according to package directions; drain and set aside.
Sprinkle tomatoes evenly with salt and pepper; sprinkle avocado with vinegar. Stir together butter and Parmesan cheese. Set aside.
Spread mayonnaise on 1 side of 4 bread slices; layer each with 1 sharp Cheddar cheese slice, 1 tomato slice, 2 avocado slices, 2 bacon slices, 1 jalapeño cheese slice, 1 tomato slice, and 1 white Cheddar cheese slice. Spread remaining 4 bread slices with dressing; place on top of sandwich with dressing side down.
Spread half of butter mixture evenly on 1 side of sandwiches. Cook 2 sandwiches, buttered side down, in a hot nonstick skillet or griddle over medium heat 2 to 3 minutes. Spread remaining butter mixture evenly on ungrilled sides of sandwiches; turn and cook 2 to 3 minutes. Repeat procedure with remaining sandwiches.
* Substitute 1/2 cup guacamole for avocado and rice vinegar, if desired.