Bring 1 cup soy sauce, ½ cup sugar, and 2 tablespoons mirin to boil in small saucepan over medium-high heat, stirring to dissolve sugar. Remove from heat and let teriyaki sauce cool completely.
Peel and thinly slice 1 (2-inch) piece of ginger. Peel 5 garlic cloves.
Combine ¾ 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 ½ cup).
Place 3 pounds chicken thighs, trimmed, into 1-gallon zipper-lock bag and add teriyaki sauce–ginger mixture. Press out air, seal bag, and turn bag to evenly coat chicken with marinade. Refrigerate for at least 1 hour or up to 24 hours.
Turn all grill burners to high, cover, and heat grill until hot, about 15 minutes. Lower to medium-high.
Place chicken on grill and cook 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 ½ inch thick and serve with reserved teriyaki sauce.
