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.
