Trim the steak of any surface fat. Cut the steak across the grain into slices ¼ inch thick, then cut the slices into 2-inch lengths. Season with salt and pepper.
Bring a large pot of salted water to boil over high heat. Add the egg noodles and stir occasionally until the water returns to a boil. Cook according to the package directions until al dente.
While the noodles are cooking, brown the steak. In a large frying pan, heat the oil over medium-high heat until it shimmers. Add the steak in batches and cook, stirring frequently, until lightly browned and still quite rare, about 1½ minutes per batch, adding more oil to the pan as needed. Transfer the beef to a plate.
When all the steak is cooked, add the mushrooms and 2 tablespoons of the butter to the frying pan and cook over medium-high heat, stirring frequently, until the mushrooms are lightly browned, about 6 minutes. Add the shallots and cook, stirring occasionally, until they soften, about 2 minutes. Remove from the heat and stir in the steak, sour cream, and the 1 tablespoon dill. Return to medium heat and simmer, stirring constantly until the sour cream is heated through, about 1 minute. Do not allow to boil. Season with salt and pepper.
When the noodles are ready, drain them and return to the cooking pot. Add the remaining 1 tablespoon butter and toss to coat. Divide the noodles among dinner plates and top each serving with the steak and sauce. Sprinkle with dill and serve.