In a medium bowl mix together the chicken, celery, onion, mayonnaise and barbeque sauce. Season with salt and pepper, to taste.
Divide the chicken salad evenly between the 4 hot dog rolls, top each with a lettuce leaf and serve.
In a small bowl add all of the spice rub ingredients. Stir together with a fork until well blended. Reserve 3 tablespoons of the rub for the Grilled Brisket recipe. Sprinkle the remaining spice rub over the chicken parts until well coated.
Preheat a grill pan over medium-high heat. Add the chicken and cook for 8 to 10 minutes per side. If using an outdoor grill, heat the grill on low. Arrange the chicken on the grill, cover and grill over indirect heat until cooked through about 10 to 15 minutes per side.
While the chicken is grilling add all the barbeque sauce ingredients to a small pan and stir well to combine. Bring to a simmer over low heat and let cook for 5 minutes. Reserve 2 tablespoons of sauce for Round 2 Recipe: Barbeque Chicken Salad.
Brush the chicken with the barbeque sauce. Cook for 3 minutes, flip and brush the other side of the chicken with remaining sauce and cook and additional 3 minutes. Let rest for 5 minutes before serving. Reserve 2 thighs for Round 2 Recipe: Barbeque Chicken Salad. Arrange the chicken on a serving platter and keep warm.
Brush the squash with the canola oil and sprinkle with salt and pepper. Put on the grill and cook for 3 to 4 minutes per side over direct heat. Remove from the grill to a cutting board and cut into 1/2-inch thick slices. Transfer to a serving bowl and serve with the chicken.