Using a fork, poke each piece of steak 10 to 12 times. Place in a large ziplock bag or shallow dish. Add soy sauce and turn until evenly coated. Cover and let marinade at room temperature for 15 minutes or in the refrigerator for up to 1 hour.
Meanwhile, place mushrooms in medium bowl, cover, and microwave until mushrooms have decreased in volume by half, 4 to 5 minutes (there should be as much as 1/4 cup of liquid in bowl). Drain mushrooms and discard liquid.
Combine mustard, water, and sugar in small bowl until smooth paste forms; set aside.
Pat steak pieces dry with paper towels and season with salt. Heat oil in 12-inch skillet over medium-high heat until just smoking.
Place steak pieces in skillet and cook until browned on all sides and meat registers 125 to 130 degrees, 6 to 9 minutes. Transfer meat to 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 dill if desired.