This is a three-alarm pizza, so have a cold drink handy.
Preheat oven to 500 degrees. On a rimmed baking sheet, stretch or roll dough into a 11-by-15-inch oval. Combine butter and 1/4 cup hot sauce; brush on dough, leaving a 1-inch border. Top with chicken. Bake until chicken is cooked through and crust is deep golden brown, 18 minutes.
Remove pizza from oven, drizzle with remaining 2 tablespoons hot sauce, and sprinkle with Monterey Jack. Bake until cheese is melted, 4 minutes more. Transfer to a cutting board and top with blue cheese, celery, and carrot.