Bring soy sauce, sugar and mirin to a boil in a small sauce pan over medium-high heat, stirring to dissolve sugar. Remove from heat and let cool completely.
Peel and thinly slice ginger. Peel garlic.
Combine 3⁄4 cup teriyaki sauce, ginger and garlic in a blender and process until smooth, about 20 seconds. Set aside the remaining teriyaki sauce for serving (should be about 1⁄2 cup).
Butterfly chicken and marinate in teriyaki/ginger mixture for at least 1 hour.
Grill chicken until lightly charred and registers 165 degrees. Transfer to a cutting board and tent with aluminum foil.
Cook broccoli and snap peas. Chop chicken. Combine and serve over rice.