Melt 1 tablespoon butter in a small heavy saucepan over moderate heat and whisk in flour, then cook roux, whisking constantly, 2 minutes. Add broth in a slow stream, whisking constantly, and bring to a boil. Reduce heat and simmer, whisking occasionally, 3 minutes. Remove from heat and keep warm. Pat beef dry and season well with salt and pepper. Heat 1/2 tablespoon butter with 1/2 tablespoon oil in a 12-inch heavy skillet over moderately high heat until foam subsides. Saute beef turning once, until browned on both sides but still pink inside, about 1 minute. Transfer to a plate with a slotted spoon.
Heat remaining 1/2 tablespoon butter with remaining 1/2 tablespoon oil in same skillet over moderately high heat until hot but not smoking, then saute shallot, stirring occasionally, until golden brown, about 3 minutes. Add mushrooms and saute, stirring occasionally, until the liquid mushrooms give off is evaporated and mushrooms are browned, about 8 to 10 minutes. Return meat with its juices to skillet, stir to combine, and then transfer to a platter.
Reheat sauce over low heat (do not let boil), then whisk in sour cream, mustard, dill, salt, and pepper. Pour sauce over beef and serve immediately.
Recipe courtesy of Gourmet Magazine