Cook the noodles according to the package directions. Meanwhile, heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the steak with 1/4 teaspoon each salt and pepper. In 2 batches, cook the steak until browned, about 1 minute per side; transfer to a plate.
Return the skillet to medium heat and add the remaining tablespoon of oil. Add the mushrooms and shallots and cook, stirring occasionally, until tender, 5 to 6 minutes. Add the wine and simmer until the liquid has reduced by half, 2 to 3 minutes.
Return the beef and any accumulated juices to the skillet and cook until heated through, 1 to 2 minutes. Serve over the noodles and top with the yogurt and dill.