Rub one teaspoon of the salt over the kidneys and let them stand for two hours. (This removes some of the strong flavor.) Remove membranes and white tubes, and cut kidneys into one-quarter-inch slices.
Sauté the kidneys and onion in three tablespoons of the butter for five minutes. Add tomatoes, celery and sweet-pepper flakes, remaining salt and the sugar. Cover and simmer for thirty minutes.
Sauté the mushrooms in remaining butter until wilted. Add them with the seasonings to the stew and cook for ten minutes longer.