Place 10 to 12 bamboo skewers in water to cover at least 30 minutes before grilling. Prepare a medium-hot fire in a charcoal grill, or preheat a gas grill to medium-high.

If using the basting sauce, in a small saucepan, combine the soy sauce, mirin, and sugar over medium-low heat and cook, stirring, until the sugar has completely dissolved and the liquid just begins to turn syrupy, 5 to 8 minutes. Remove from the heat and set aside in a small bowl.

Drain the skewers. Thread the chicken pieces alternately with the onion pieces onto the skewers, using about 3 pieces of chicken and 2 pieces of onion for each skewer. As each skewer is ready, place on a plate. Brush the chicken and onion pieces lightly on all sides with the oil.

The grilling goes quickly, so set up your workstation next to the grill before you begin: the plate holding the skewers, the bowl of sauce and a basting brush and/or the salt, and tongs for turning the skewers. If using the basting sauce, either dip each skewer in the sauce or brush the sauce on the skewers and then place directly over the heat. Grill the skewers, turning 3 or 4 times and basting with additional sauce, until the chicken is cooked through and the onions are scorched around the edges, about 5 minutes total. If the sauce appears to be burning at any point, move the skewers to a cooler part of the grill, or reduce the heat if using a gas grill. If using salt instead of the basting sauce, place the skewers directly over the heat and grill, turning 3 or 4 times, until cooked through, about 5 minutes total. Sprinkle both sides of each skewer with salt and remove from the grill.

Serve the skewers piping hot.