Preheat oven to 475 degrees. Place chicken, skin side up, on a rimmed baking sheet and season with salt, pepper, and 1/2 teaspoon cayenne. Cook until skin is golden and crisp and chicken is cooked through, 20 minutes, rotating sheet halfway through. Meanwhile, in a large bowl, whisk together mayonnaise, 2 tablespoons vinegar, and half the blue cheese. Add celery and romaine, season with salt and pepper, and toss to combine. Sprinkle salad with remaining blue cheese.
In a small pot, melt butter with 3/4 teaspoon cayenne over medium-low. Whisk in 3 tablespoons vinegar and remove from heat; season with salt and pepper. In a medium bowl, toss chicken with butter mixture to coat. Divide chicken among four plates, drizzle with any extra butter mixture, and serve with salad.