Food clings well to both single- or double-pronged bamboo or wooden skewers, so you can use either for this recipe. They must both be soaked in cool water for half an hour before grilling to avoid catching fire.
Soak 28 seven-inch bamboo skewers in water 30 minutes; drain. Meanwhile, combine soy sauce, mirin, sake, sugar, ginger, and garlic in a small saucepan. Bring to a boil over medium-high and cook, stirring occasionally, until reduced to 2/3 cup, 8 to 10 minutes. Stir in 1/4 teaspoon pepper; let cool.
Thread each chicken wing onto a skewer. Thread 2 pieces each eggplant and scallion onto skewers. Thread 2 pieces each shiitake and scallion onto skewers. Thread shishitos 4 per skewer. Divide enokis into 12 equal bundles; wrap each in a piece of bacon, then thread 3 per skewer. Thread asparagus 5 per skewer. Lightly brush eggplant, shiitake, and asparagus skewers with oil.
Preheat grill for direct-heat cooking (medium-high on a gas grill). Lightly brush grates with oil. Grill chicken skewers 5 minutes. Flip chicken and add eggplant and shiitake skewers; grill 5 minutes more. Flip chicken, eggplant, and shiitake skewers, and add shishito, bacon-wrapped enoki, and asparagus skewers; continue to grill until chicken is just cooked through, eggplant and shiitakes are tender, bacon is crisp and browned, and shishitos and asparagus are crisp-tender, 5 to 7 minutes more. In the last minute of cooking, brush chicken, eggplant, and shiitake skewers with soy mixture a few times while flipping repeatedly.
Remove skewers from grill. Brush chicken, eggplant, and shiitake skewers with more soy mixture, then sprinkle with sesame seeds. Sprinkle shishito, enoki, and asparagus skewers with flaky salt. Serve, with lemon wedges on the side.