Place chicken into a large pot with enough water to cover; bring to a boil. Cook until chicken comes off the bone easily, about 3 hours.
While the chicken is cooking, combine ketchup, mustard, brown sugar, garlic, honey, steak sauce, lemon juice, and liquid smoke in a saucepan. Stir to combine, then season with salt and pepper. Bring mixture to a boil, then reduce the heat and simmer for about 10 minutes. Remove from the heat and set aside for flavors to mingle until chicken is ready.
Transfer cooked chicken to a cutting board. Remove meat and discard skin and bones. Chop or shred meat into small pieces.
Add chicken to sauce and stir to combine. Heat over medium low until BBQ flavor soaks into chicken, about 15 minutes.
Spoon chicken onto buns and top with coleslaw.
BBQ Chicken Sandwiches · Dinner Match Lab | Dinner Match Lab