Mix together the shredded chicken, monterey jack cheese, cheddar cheese, and hot sauce in a large bowl.
Place a wonton wrapper on a flat surface, and spoon 1 tablespoon of the buffalo chicken mixture into the center of the wonton. Lightly moisten the edge of the wonton wrapper with water, and fold 1 corner diagonally across to the other corner to create a triangle. Press the wrapper together at the seam to seal. Lightly brush the wonton with egg white, and dip the wonton into breadcrumbs on both sides to coat. Repeat with the remaining wontons.
Heat oil in a heavy-bottomed dutch oven or electric deep fryer to 350ºF. Fry wontons in batches for about 2 to 3 minutes, or until the outside is crispy and golden brown. Place wontons on a paper-towel-lined plate to drain off any excess oil, and serve warm with celery, buffalo wing sauce, and blue cheese dressing.