Mix cream cheese with 2 tablespoons buffalo sauce until smooth and well combined in a medium bowl.
Toss shredded chicken with remaining buffalo sauce in a separate bowl. Taste and adjust heat level to preference.
Spread butter on one side of each bread slice. Flip four slices and spread cream cheese mixture on unbuttered sides.
Layer buffalo chicken, Monterey Jack, and cheddar cheese over cream cheese. Add blue cheese crumbles if using.
Top with remaining bread slices, buttered side facing out. Press gently to compact.
Heat skillet over medium heat. Cook sandwiches 3-4 minutes per side until golden brown and cheese melts.
Cover pan for last minute to ensure cheese melts completely. Rest 2 minutes before cutting.
Serve immediately with celery sticks and extra buffalo sauce for dipping.