Place 1/4 cup flour on a large plate and dredge pork slices in it, shaking off the excess.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add the pork and cook until golden, crispy and just barely pink in the center, 2 to 3 minutes per side. Transfer the pork to a plate; tent with foil to keep warm.
Heat the remaining 1 tablespoon oil in the skillet over medium-high heat. Add shiitake and white mushrooms and cook, stirring often, until softened and beginning to brown, 5 to 7 minutes. Sprinkle the remaining 2 tablespoons flour over the mushrooms; stir to coat. Add broth, sake, vinegar and pepper and bring to a simmer over high heat, stirring often. Reduce heat and continue simmering, stirring occasionally, until the sauce is thickened and slightly reduced, 4 to 6 minutes. Stir in scallions and miso. Return the pork to the pan, turn to coat with the sauce, and simmer until heated through, about 1 minute. Serve the pork topped with the sauce.
Ingredient notes: Sake is a dry rice wine generally available where wines are sold. Junmai, a special designation for sake, denotes sake brewed from rice that has been milled less than other special-designation sakes. More pure than other sakes, junmai has no distilled alcohol added. It is characterized by a well-rounded, rich flavor and body and more acidity than most sakes.Miso is fermented bean paste made from barley, rice or soybeans; it is used in Japanese cooking to add flavor to dishes such as soups, sauces and salad dressings. A little goes a long way because of its concentrated, salty taste. Miso is available in different colors, depending on the type of grain or bean and how long it's been fermented. In general, the lighter the color, the more mild the flavor. It will keep, in the refrigerator, for more than a year.