Heat grill to medium. Whisk together the mirin, sake, soy sauce, pineapple juice, 6 tablespoons sugar or honey, and ginger in a medium saucepan and cook on the grates of the grill until the sugar has melted and the mixture has thickened slightly, about 15 minutes. Remove from the heat and let the sauce cool completely.
Whisk together the rice vinegar, 1 tablespoon sugar, red pepper flakes, and 1/2 teaspoon salt in a medium bowl. Add the cucumbers and stir to combine. Cover and let marinate in the refrigerator for at least 1 hour.
Heat grill to medium. Lightly season the chicken with salt and pepper. Place the chicken on the grill, skin side down, and grill until golden brown and slightly charred, about 4 to 5 minutes. Brush with some of the teriyaki sauce, turn over, and continue grilling and brushing with the sauce every few minutes until cooked through, about 25 more minutes. Transfer to a platter with the marinated cucumbers and serve.