Fire up the grill or a grill pan over medium heat.
In a medium bowl, combine the chicken, bread, 1/4 cup of milk, onion, pepper, and salt.
Fold together, allowing the bread to absorb all of the liquid. Do not mash. Keep the mixture loose and irregular.
Divide the chicken mixture into 6 equal portions. Form 6 (1/2-inch-thick) burgers, bringing them together just tightly enough to hold their shape.
Combine 1/4 cup hot sauce and butter in a shallow dish. Working with 1 patty at time, roll patties in butter mixture.
In a medium bowl, combine 1 tablespoon of the milk, 1 tablespoon hot sauce, the blue cheese, the buttermilk, and the mayonnaise. Add the carrots and celery, and fold to combine. Let wilt slightly at room temperature while the burgers are grilling.
Grill the burgers 5 minutes per side, or until the internal temperature reaches 155°. They'll carry over.
Platter the burgers, brushing each with 1/3 cup celery salad. Sprinkle burgers evenly with green onions.