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Heat 2 teaspoons oil in a large skillet over high heat until shimmering but not smoking. Add steak and cook until browned on both sides, 3 to 4 minutes per side. (The meat will be rare, but will continue to cook as it rests.) Transfer to a cutting board and let rest for 5 minutes. Cut lengthwise into 2 long pieces then crosswise, across the grain, into 1/4-inch-thick slices.
Heat the remaining 1 tablespoon oil in the pan over medium heat. Add mushrooms, onion, thyme, salt and pepper and cook, stirring often, until the vegetables are very tender and lightly browned, 8 to 12 minutes. Sprinkle flour over the vegetables; stir to coat. Stir in broth, cognac (or brandy) and vinegar and bring to a boil, stirring often. Reduce heat to a simmer, and continue cooking, stirring often, until the mixture is thickened, about 3 minutes. Stir in sour cream, chives (or parsley), the sliced steak and any accumulated juices. Bring to a simmer and cook, stirring, until heated through, 1 to 2 minutes more.