Set up your grill for indirect grilling and heat to medium-high. If enhancing a charcoal fire, add the wood chunks or chips to the coals; if enhancing a gas fire, place the chunks or chips in your grill’s smoker box or place chunks under the grate directly over one or more burners.

Cut the large squash lengthwise into 6 wedges. Cut the small squashes lengthwise into quarters. Scoop out and discard the strings and seeds.

Make the sauce: Pour the soy sauce, sake, sesame oil, maple syrup, and chili sauce into the foil pan and whisk to mix. Gently flatten the scallion white, ginger, and garlic with the side of a cleaver or chef’s knife. Stir them into the sauce. Add the squash pieces, turning each several times to coat all sides.

Place the squash, cut sides down, in its foil pan on the grill grate away from the heat. Indirect grill the squash until almost tender, about 30 to 45 minutes, turning the pieces with tongs a couple of times so they cook evenly.

Brush or scrape the grill grate clean (the portion directly over the fire) and oil it well. Using tongs, lift the squash pieces out of the pan with the teriyaki sauce and arrange directly over one of the fires, cut sides down. Move the foil pan with the sauce over the other fire.

Direct grill the squash pieces until sizzling and darkly browned on all sides (the two cut sides and the skin side), 1 to 2 minutes per side, 3 to 6 minutes in all. If you like, give each a piece a quarter turn after 1 minute to lay on a crosshatch of grill marks. Meanwhile, boil the teriyaki sauce in its pan until thick, brushing it on the squash pieces as they grill.

Transfer the squash to a platter or plates. Spoon any remaining teriyaki sauce over them. (I leave the ginger and garlic in, but you can certainly discard them if you are so inclined.) Sprinkle with the scallion greens and sesame seeds (if using), and dig in!