For the burgers: Melt 9 tablespoons of the butter in a pan, then pour it into a measuring cup.
Put the ground chicken in a bowl and add 1 tablespoon of the melted butter, 1 tablespoon of the hot sauce and the salt and pepper. Mix and form into 2 patties.
Set a nonstick pan over medium-low heat and add the oil to the pan. Add the patties to the pan, turn the heat up to medium and cook until they are browned on the first side, about 4 minutes.
Meanwhile, mix the remaining 8 tablespoons hot sauce into the remaining butter in the measuring cup.
Flip the patties and put a couple tablespoons of the butter and hot sauce mixture on each. Cook another 4 minutes.
For the blue cheese dressing: Meanwhile, mix the mayonnaise, sour cream, blue cheese crumbles, vinegar and some salt and pepper in a bowl. Taste and adjust with more vinegar, salt and pepper if needed.
Add the remaining 2 tablespoons butter to a skillet and heat over medium-low heat until the butter is melted and the skillet is hot. Add the onion rolls cut-side down and toast until golden, about 2 minutes.
Flip the patties again, put another couple tablespoons of the butter and hot sauce mixture on each patty and cook until the patties are cooked through, about 2 more minutes.
Spread a little of the butter and hot sauce mixture on the top and bottom buns. Top each bottom bun with a patty, then some sliced celery, blue cheese dressing, blue cheese crumbles, celery leaves and the top buns. Serve with more blue cheese dressing and the celery sticks.