Prepare the beef. Trim the rib-eye as necessary and pound it to an even thickness of roughly 1/3 inch. Cut the steak into strips and season the pieces generously on both sides with kosher salt and freshly ground black pepper.
Sear the first batch. Place a large skillet over high heat and add 1 tablespoon of the neutral oil. Once the pan is hot, arrange half of the seasoned steak strips in the skillet without crowding them. Sear for approximately 30 seconds on each side, allowing the exterior to brown while leaving the center undercooked. Transfer the beef to a plate.
Sear the remaining beef. Pour the remaining 1 tablespoon of oil into the skillet and repeat the searing process with the second batch of steak. Transfer the browned strips to the same plate, retaining any juices that collect there.
Cook the onion and mushrooms. Reduce the heat to medium and melt the unsalted butter in the same skillet. Add the sliced onion and cook for approximately 1 minute, until it begins to soften. Stir in the cremini mushrooms and continue cooking for about 4 minutes, stirring occasionally. The mushrooms should become golden as their released liquid cooks away and is reabsorbed.
Form the sauce base. Scatter the all-purpose flour evenly over the mushroom and onion mixture. Stir thoroughly and cook for approximately 1 minute so the flour loses its raw flavor.
Add the broth and mustard. Pour 1/2 cup of the beef broth into the skillet while scraping the browned residue from the bottom of the pan. Stir until a smooth, lightly thickened mixture forms. Add the remaining 1 1/2 cups beef broth along with the Dijon mustard.
Reduce the sauce. Bring the mixture to a gentle simmer and allow it to cook for approximately 5 minutes, or until the sauce has reduced slightly.
Incorporate the sour cream. Remove the skillet from the heat. Stir in the room-temperature sour cream until completely blended and the sauce has a smooth, creamy consistency. Keep the pan off direct heat during this stage to help prevent the sour cream from curdling.
Return the beef to the skillet. Add the seared steak strips and all accumulated juices back to the finished sauce. Place the skillet over low heat and stir gently for approximately 1 minute, heating the beef only until warmed through.
Serve. Spoon the beef stroganoff over cooked egg noodles or tagliatelle. Finish each portion with a scattering of freshly chopped chives.