Heat the oil to 375 degrees and fry the potato cubes until browned. Drain and keep warm in a 200-degree oven.
Season the steaks with salt and pepper. Heat 1 Tbs. butter in a heavy skillet over medium-high heat and broil the steaks, three at a time. They will stick to the pan, then release somewhat. At that point turn them with a metal spatula (not a fork) and cook the other side to the preferred degree of doneness. Add the remaining butter and broil the remaining steaks. Remove from the pan and keep warm in the oven.
Lower the heat under the skillet to medium-low. Add the wine to the pan and whisk to dissolve the pan juices and browned bits. Let half the wine boil away, then add the ham, green onions, mushrooms, Worcestershire, and Tabasco. Cook until the mushrooms are tender--about three minutes. Lower the heat as low as it will go.
Add the fried potatoes, and toss with the other ingredients to distribute. Season with salt and pepper to taste.
Place a filet mignon on each warm serving plate. Scatter the pan contents around the steak. Spoon about 3 Tbs. bearnaise sauce over everything and serve immediately.