Combine dashi, soy sauce, sake, sugar, and mirin in a small saucepan or donburi pan and bring to a simmer over medium heat. If using onion, add to broth and simmer until tender, about 5 minutes. Add sliced fried cutlet and let simmer for 1 minute. Meanwhile, beat together eggs and scallions in a small bowl. Pour egg mixture on top of cutlet and around broth. Cover and cook until eggs are as set as you'd like them, about 1 minute for very soft or 2 minutes for medium. Slide broth, egg, and chicken out on top of a bowl of rice. Sprinkle with scallions and serve.