For the short ribs: Stir short ribs, soy sauce, mirin, garlic powder, garlic, thyme and 4 cups cold water together in a pot over high heat and bring to a boil. Turn down to a simmer and cover. Cook until the meat is very tender, about 4 hours.
Strain the short ribs from the liquid and put the meat in a mixing bowl. Before it cools, grab tongs and shred it. Return the meat to the liquid.
For the pickled red onions: Meanwhile, combine vinegar, salt, sugar and 3 cups water in a pot over high heat. Put onions in a heat-safe container. Once the pickling liquid comes to a rolling boil, turn off the heat and pour it over the onions. Make sure you use a container that will cover the red onions completely with the liquid; weigh down the onions with a plate if necessary. Let it sit at room temp so the color bleeds out from the red onions, about 30 minutes. Refrigerate. (Pickled onions can be used as soon as they are cold, but they are best after at least 3 days.)
For the grilled cheese: Preheat a toaster oven or regular oven to 350 degrees F.
Spread butter on both sides of the slices of bread, then put down mozzarella on one side and Cheddar on the other. Bake until the cheese melts. Meanwhile, heat a saute pan over medium-high heat.
Add the short rib and some pickled red onions to the bread and sandwich them together. Add a bit of oil to the pan, then put the sandwich in the pan. Grill until crispy golden brown on both sides. Repeat with remaining ingredients.