Rinse, drain, and discard stems from 4 fresh shiitake mushrooms (2-in. caps). Finely chop caps and put in a 6- to 8-inch frying pan with 2 tablespoons water, 1 tablespoon salad oil, and 1 tablespoon soy sauce. Stir over high heat until liquid evaporates, 3 to 4 minutes. Scrape into a large bowl. In a 2- to 3-quart pan, rinse 1 1/2 cups short-grain white rice with cold water until water is clear. Drain. Add 1 1/2 cups water, cover, and bring to a boil over high heat. Reduce heat to low and cook until water is absorbed, about 15 minutes. Dump rice into bowl with shiitake mixture. Sprinkle with 5 to 6 tablespoons seasoned rice vinegar. Mix with a spatula and let stand about 10 minutes. To rice add 4 teaspoons finely chopped sliced pickled ginger, 1/4 cup diced red bell pepper, and 3/4 cup diced Japanese or English cucumber. Mix.