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.