Chunky bleu cheese dressing, as desired for dipping
Heat peanut oil in dutch oven (no more than half-full to avoid bubbling over!) over medium-high heat until it reaches 350°F (177°C), monitoring the temperature with your Thermometer.
Pat chicken wings completely dry with paper towels.
Using a slotted spoon or spider, add some of the chicken wings into hot oil and fry for 7-10 minutes. Be sure not to crowd the pan. Adding too many wings at once will dramatically drop the temperature of your oil and risk a boil-over. You’ll need to fry in 2-3 batches.
While the wings are cooking, heat the hot sauce in a small pot over medium-low heat, checking it with your Thermometer until it reaches 110°F (43°C). Remove from heat.
Whisk one or two pieces of butter into the hot sauce until completely melted.
Add the rest of the butter and whisk until all the pieces have melted. Set aside.
When your 7-minute timer goes off, quickly spot-check the internal temperature of a few chicken wings with an instant-read digital thermometer like ThermoPro TP19, to make sure it has reached 165°F (74°C). Remove from the hot oil with a slotted spoon or spider and transfer to a sheet pan lined with paper towels or a cookie sheet with a cooling rack to drain excess oil. The wings will be crisp on the outside, juicy on the inside.
Allow the oil temperature to come back up to 350°F (177°C) before adding more wings.
After all of the wings are cooked and drained, place them in a large bowl, add hot sauce, and toss to coat. Grab plenty of napkins, serve with bleu cheese dressing and celery sticks.