Preheat your grill to medium-high heat and pat your chicken dry with a paper towel.
In a large bowl, toss the chicken wings with olive oil, salt, black pepper, and Old Bay seasoning. Ensure the wings are evenly coated with the seasoning mixture.
Place the wings on the preheated grill and cook for about 10-12 minutes per side, or until they reach an internal temperature of 165°F (74°C). Keep an eye on them to avoid burning and flip them halfway through to cook them evenly. I would recommending starting on a lower heat, to cook the wings inside, and then bring up the heat to get a nice char and grill on them. Flip as needed.
While the wings are grilling, prepare the sauce. In a small saucepan, melt butter over medium heat. Allow it to continue cooking until it turns a rich golden brown color, swirling occasionally to ensure even browning, about 5-6 minutes. Keep mixing to ensure the butter does not burn, and that it just browns and toasts.
Once the butter has browned, remove the saucepan from the heat and add the hot sauce, salt, pepper, and Old Bay seasoning. Stir well to combine all the ingredients.
Transfer the grilled wings to a large bowl and pour your sauce over them. Toss the wings gently, ensuring each one is coated in the sauce.
Serve your grilled wings with celery sticks, carrot sticks, and your favorite dipping sauce (like ranch dressing, or blue cheese dressing!)