In a medium saucepan, combine mirin, sake, 1/2 cup sugar, the ginger, garlic, and soy sauce. Bring to a boil and cook until sugar has dissolved and sauce has thickened slightly, about 2 minutes. To cool quickly, pour into a heatproof 9- by 13-in. pan and refrigerate until cool. Add chicken and toss to coat. Chill and marinate at least 30 minutes and up to 24 hours.
Heat a grill to medium-high (450°). Transfer chicken to a platter. Pour marinade into a small saucepan and boil to make a syrupy sauce, about 8 minutes. Discard ginger and garlic.
In a large bowl, whisk together vinegar, salt, and 1 tsp. sugar. Add cucumber and onion; toss to coat.
Grill chicken, turning once, until no longer pink in center (cut to test), about 2 minutes per side. Coat chicken all over with reduced marinade.
Add avocado to salad and sprinkle with sesame seeds. Serve chicken with salad and remaining sauce, for drizzling.