Step 1

Combine soy sauce, water, mirin, ginger, and garlic in saucepan. Toss chicken with 1/2 cup soy sauce mixture in large bowl and refrigerate for at least 30 minutes or up to 1 day. Meanwhile, whisk sugar and cornstarch into remaining soy sauce mixture in saucepan. Simmer over medium-high heat until thickened and reduced to 3/4 cup, about 5 minutes. (Glaze can be refrigerated for up to 1 day.)

Step 2

Heat all burners on high for 15 minutes, then turn all burners to medium-low. (For charcoal grill, light 50 coals; when covered with fine gray ash, spread in even layer. Set cooking grate in place and heat, covered, for 5 minutes.) Scrape and oil cooking grate. Grill chicken skin-side up, covered, until well browned on bottom and internal temperature of meat registers 120 degrees, 15 to 20 minutes. Flip chicken skin-side down and grill, covered, until skin is rendered and deep brown and meat registers 160 degrees, 15 to 20 minutes. Brush chicken with 1/4 cup glaze and grill, covered, until glaze is caramelized, about 5 minutes.

Step 3

Transfer chicken to serving platter and brush with 1/4 cup glaze. Let rest 10 minutes. Serve, passing remaining glaze at table.