Melt the butter in a 12-inch heavy-duty skillet over medium-high heat. Add the mushrooms, season generously with salt, and cook, stirring occasionally, until golden, about 8 minutes. Reduce the heat to medium, add the shallot, and cook until softened, about 3 minutes more. Remove the skillet from the heat and carefully add the Cognac, stirring to deglaze the pan. Add the broth, bring to a boil over medium heat, reduce the heat to low, and simmer for 3 minutes. Cover the skillet, and set aside.
Meanwhile, heat the oil in a 12-inch cast-iron skillet over medium-high heat. Pat the steaks dry and season generously on both sides with salt and pepper. When the pan is very hot, add the steaks. Sear until a dark crust forms, 4 to 5 minutes. Turn the steaks, and cook to desired doneness, about 5 minutes more for medium rare. Remove the skillet from the heat, transfer the steaks to a platter, tent with foil, and let rest for 5 minutes.
With a rubber spatula, scrape the mushroom mixture into the cast-iron skillet. Add the sour cream and mustard, stirring until heated through and well mixed. If necessary, return the skillet to the stovetop to heat the sauce. Stir in the dill.
Transfer the steaks to plates, spoon the sauce over, and serve.
Button and cremini mushrooms, shown in photo, are delicious in this dish, but so are other varieties. Trumpet-shaped chanterelles have an oaky flavor while maitakes, also called hen of the woods, are rich and earthy.