Combine hot sauce and cayenne pepper in a small saucepan over medium heat. Bring just to a simmer. Gradually whisk in butter, a few pieces at a time, until combined. Reduce heat, and keep warm until ready to use.
Place a wire rack in the bottom of a heavy-bottomed stockpot. Add enough oil to come 4 inches up the side, and heat over medium-high heat until it reaches 400 degrees on a deep-fry thermometer. Line a rimmed baking sheet with a wire rack; set aside.
Working in batches, carefully add chicken to heated oil and cook until dark golden brown, 15 to 20 minutes. Transfer to wire rack on baking sheet to drain.
Place mayonnaise, sour cream, lemon juice, and blue cheese in a bowl, and stir to combine. Season with salt and pepper.
Transfer cooked wings to a large, wide bowl. Add hot sauce mixture, and toss to coat. Serve wings with blue cheese sauce and celery sticks.