Carefully cut the fillets into thin strips, salt them just a little and roll them; set aside.
In a saucepan, melt the butter over medium heat and add the onions. When the onions are light golden, add the Champagne. Before it boils add the fillets, and cook until they are opaque, then remove the fillets from the pan.
Cook the sauce until it's reduced by half. Add the mushrooms, the heavy cream and a pinch of salt, and cook another couple of minutes to reduce it more. Cover the fillets with the sauce and serve.