Prepare a grill for indirect heat: For gas, preheat to medium high, then turn off the burners on one side and turn the other burners to medium low. For charcoal, once the coals ash over, push them to one side.
Mix 2 teaspoons vegetable oil, the brown sugar, 1 teaspoon five-spice powder, 1 teaspoon salt and 1/4 teaspoon pepper in a bowl. Rub on the chicken. Grill skin-side down over direct heat until marked, about 5 minutes. Flip and move to the cooler side of the grill; cover and cook until a thermometer registers 165 degrees F in the thigh, 30 to 40 minutes.
Meanwhile, heat the remaining 2 teaspoons vegetable oil in a saucepan over medium heat. Add 3 teaspoons ginger, the garlic and the remaining 1/2 teaspoon five-spice powder and cook 2 minutes. Add the hoisin sauce, 1 tablespoon vinegar and 3 teaspoons honey; cook 1 minute. Remove from the heat and add 1 teaspoon sesame oil.
Toss the cucumbers with the remaining 1 tablespoon vinegar, 1/2 teaspoon honey, 1/4 teaspoon each sesame oil and ginger, and salt to taste. Top the chicken with the sauce. Serve with the cucumbers.