Using a fork, poke each piece of steak 10 to 12 times. In a large zip-top bag or shallow dish, add poked steak and soy sauce and turn until evenly coated. Cover and marinate at room temperature for 15 minutes or in the refrigerator for up to 1 hour.
Meanwhile, in a medium bowl, add mushrooms. Cover and microwave until mushrooms have decreased in volume by half, 5 to 10 minutes (there should be as much as 1/4 cup of liquid in bowl). Drain mushrooms and discard liquid.
In a small bowl, whisk together mustard, water, and sugar until a smooth paste forms. Set aside.
Pat steak pieces dry with paper towels and sprinkle with salt. In a large (12-inch of larger) skillet, heat oil over medium-high heat until just smoking.
Add steak pieces to skillet and cook until browned on all sides and the meat registers 125 to 130 degrees, 6 to 9 minutes. Transfer to a large plate and tent with foil.
To the skillet, add mushrooms, onion, and 1/2 teaspoon salt. Cook until vegetables begin to brown and dark bits form on bottom of pan, 6 to 8 minutes.
Add flour and tomato paste and cook, stirring constantly, until onions and mushrooms are coated, about 1 minute. Stir in broth and mustard paste and bring to simmer, scraping bottom of pan to loosen browned bits. Reduce heat to medium and cook until sauce has reduced slightly and begun to thicken, 4 to 6 minutes.
While sauce is reducing, slice steak pieces against grain into 1/4-inch thick slices. Stir meat and any accumulated juices into thickened sauce and cook until beef has warmed through, 1 to 2 minutes.
Remove pan from heat and let any bubbles subside. Stir in sour cream and season to taste with salt and pepper. Serve over hot buttered noodles and garnish with chives if desired.