Add the hot water, brown sugar, and espresso to a medium mixing bowl and stir to dissolve the sugar and espresso powder.
Add the ketchup, mustard, vinegar, liquid smoke, and soy sauce to the bowl and whisk well to combine. Remove half of the sauce to a small bowl to serve as a dip for the chicken.
Pound or slice the chicken breasts into an even thickness so they cook evenly. Place the chicken on the grill and brush the tops with the sauce.
Cover the grill and cook for 5 minutes.
Flip the chicken and brush with the sauce. Continue cooking for 2 minutes and then flip and baste again. Repeat, basting and flipping every 2 minutes, until the chicken is cooked through.
Sprinkle with parsely before serving. Serve with the reserved dipping sauce.
The espresso powder does not make this taste like coffee, but does add amazing flavor to this. It's optional, but highly recommended. Hot brewed coffee in place of the water and espresso powder may be substituted.