Yoshinoya, established in 1899, is the largest chain among beef bowl (or gyudon) restaurants and one of the leading fast food chains in Japan. I first had their Beef Bowl in one of their California restaurants and loved it. Cheap, Quick and Yummy.
Slice onion thinly. Cut thinly sliced beef into bite-sized pieces. Set aside. Put dashi, soy sauce, sugar, mirin, sake, ginger juice and garlic in a pan. Add onion slices in the pot and bring to just under a boil then lower to a simmer. Allow to simmer for minimum of 5 minutes.
Boil enough water for 1/2 package of frozen carrots, broccoli and cauliflower. When water is at a boil, add contents of package of vegetables as well as 1/4 head of cabbage. When vegetables are almost done, (4-5 minutes), drain them. Add beef and drained vegetables into the saucepan and simmer for 5 minutes. If veggies cooked earlier then add additional time to allow sauce to return to simmer and fully cook meat.
Serve hot steamed rice in a deep rice bowl. Put the beef and vegetable topping on the top of rice. Place some red ginger on the top if you would like.