Soak 12 bamboo skewers in cold water for 30 minutes.
In a small saucepan, combine roughly equal amounts of sake, soy sauce, mirin, and sugar. Bring to a boil on medium-high heat, then reduce to medium heat and simmer for 10 minutes. Mix starch with 1 teaspoon of water, then stir this mixture into the sauce. Transfer the sauce to a small bowl.
Cut the chicken into bite-sized pieces.
Cut the onion diagonally into 3 cm pieces.
Preheat grill to high heat. Thread chicken pieces and onion pieces onto the soaked skewers. Place on the grill and cook for 5 to 10 minutes, turning every 1 to 2 minutes and basting with sauce. Chicken is done when it reaches 74°C (165°F) internally.
Serve the kebabs with the remaining yakitori sauce.