Bring soy sauce, sugar and mirin to a boil in a small saucepan over medium-high heat, stirring to dissolve the sugar. Remove from heat and let teriyaki sauce cool completely.
Combine 3/4 cup teriyaki sauce, ginger and garlic in blender and process until smooth, about 20 seconds. Set aside remaining teriyaki sauce for serving (you should have about 1/2 cup).
Place chicken in 1-gallon zipper-lock bag and add teriyaki sauce-ginger mixture. Press out air, seal bag and turn to coat chicken with marinade. Refrigerate for at least 1 hour or up to 24 hours.
Turn on all burners to high, cover, and heat grill until hot, about 15 minutes. Turn all burners to medium-high.
Clean and oil cooking grate. Place chicken on grill and cook (covered if using gas) until chicken is lightly charred and registers at least 175 degrees, 6 to 8 minutes per side, rearranging as needed to ensure even browning. Transfer to cutting board, tent with aluminum foil, and let rest for 5 minutes. Slice crosswise 1/2 inch thick and serve with reserved teriyaki sauce.