Add 1 tablespoon melted butter to a skillet; saute fillets for 4 minutes on each side (actually you need to saute fillets until desired degree of doneness; 4 minutes/side is fairly rare which is how my husband likes them) remove from skillet and keep warm; drain and discard pan drippings.
Add 1 tablespoon melted butter to skillet over low heat.
Saute/stir mushrooms and shallots for 2-3 minutes.
Add in 1/2 cup Burgundy wine; cook over high heat until wine is reduced by half.
Mix together 1 teaspoon butter and 1 teaspoon flour to make a paste.
Add to mushroom mixture; cook and stir constantly for 30 seconds.
Add the last tablespoon butter and 2 tablespoons Burgundy; stir to combine.
Serve fillets on individual plate and spoon sauce over the top.