Pulse sour cream, blue cheese, mayonnaise, garlic, milk and lemon juice in a food processor until combined but still chunky. Season with salt and pepper. Cover and refrigerate until ready to serve.
Preheat the oven to 425 degrees Fahrenheit.
Coat a rimmed baking sheet with cooking spray.
In a deep skillet, add chicken, 3 pieces of celery, 3 pieces of carrot, bay leaves, chicken broth, hot sauce, paprika, butter, honey and 1/2 cup water and bring to a simmer over medium heat. Cook until chicken is almost cooked through, about 18 minutes.
Remove chicken and transfer to baking sheet. Bake until the skin is golden and crisp, about 30 to 35 minutes. (Turn chicken over after first 15 minutes.)
Continue to simmer sauce in pan until slightly thickened, about 15 minutes. Discard the celery, carrot and bay leaves. Reserve one-quarter of the sauce for serving, set aside.
After baking, return chicken to skillet and toss with sauce once more to coat.
Place chicken on a serving platter, with sides of prepared blue cheese dip, remaining sauce, celery and carrot sticks for garnish. Serve immediately. Refrigerate any leftovers.