Place oven rack in lower middle position and preheat to 250°F Meanwhile, stir together flour, salt, and black pepper on a large plate. Dredge both sides of each cube steak in flour mixture, shaking off excess.
Add one tablespoon each of oil and butter to Dutch oven and place over medium high heat. When oil is shimmering, add two cube steaks. Cook until golden brown, about four minutes on each side. Remove cube steaks to a plate and set aside. Repeat with remaining two cube steaks, adding more oil and butter if necessary.
Carefully pour out any fat left in Dutch oven, then add remaining butter and oil. Turn heat to medium-high, and when oil shimmers, add onion, garlic, and mushrooms. Cook, stirring frequently, until onion is caramelized, about three minutes. Add wine. Using wooden spatula, scrape up any browned bits from bottom of pan.
Add Worcestershire sauce, mustard, and stock. Stir well, then add cube steaks, nestling them into sauce as much as possible. Cover dutch oven and place on lower-middle rack of oven. Cook until very tender, about 25 minutes. Serve with mashed potatoes.