Choose a sauté pan that can subsequently accommodate all the steaks without overlapping. Put in the olive oil and sliced onion, and turn on the heat to medium.

Cook until the onion becomes translucent, then dredge the steaks on both sides in the flour, and put them in the pan, turning the heat up to high. Cook the meat about 1 minute on each side, sprinkle it with salt and a few grindings of pepper, and transfer it with a slotted spoon to a warm plate.

Add the cut-up reconstituted mushrooms and the filtered water from their soak to the pan, turn the heat down to medium, and cook until the mushroom liquid has completely evaporated. Stir from time to time.

Add the wine and let it simmer away for a minute or so, stirring frequently, then Put in the tomatoes with their juice and adjust heat to cook at a slow, but steady simmer. Add salt and pepper, correcting seasoning to taste, and stir from time to time. Cook for about 10 or 15 minutes, until the oil floats free from the tomatoes.

Turn the heat up to high, return the steaks to the pan together with any juices they may have shed in the plate, and turn them once or twice to reheat them in the sauce, but for no longer than 20 or 30 seconds. Transfer the meat and the entire contents of the pan to a warm platter and serve at once.