1. 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.
2. 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.
3. In a large bowl, whisk together vinegar,
salt, and 1 tsp. sugar. Add cucumber and onion;
toss to coat.
4. 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.
5. Add avocado to salad and sprinkle with
sesame seeds. Serve chicken with salad and
remaining sauce, for drizzling.
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