Steak Stroganoff

8 ounces extra-wide egg noodles

3 medium red onions, divided

2 tablespoons olive oil, divided

1 tablespoon sugar

1 tablespoon red wine vinegar

1 1/2 teaspoons salt, divided

1/2 teaspoon ground black pepper, divided

2 tablespoons unsalted butter

12 ounces sliced mushrooms

1/4 teaspoon dried thyme

2 tablespoons all-purpose flour

2 cups unsalted chicken or beef broth

1/3 cup heavy cream

1/2 cup sour cream

1 tablespoon chopped fresh parsley

(1 1/2-pound) top sirloin steak, about 1-inch thick

1. Bring a large pot of salted water to a boil. Add the noodles and cook according to package directions; drain and reserve.
2. Thinly slice 2 of the onions. Heat 1 tablespoon of the oil in a 10-inch nonstick skillet over medium heat. Add the sliced onions and sugar; cook, stirring occasionally, until softened (but not browned), about 11 to 12 minutes. Stir in the vinegar and cook 1 minute, stirring. Remove from the heat and season with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper.
3. Chop the remaining onion in small (1/4-inch) pieces. Melt the butter in a 12-inch nonstick skillet over medium-high heat. Add the onion, mushrooms, thyme, and 1/4 teaspoon of the salt. Cook, stirring occasionally, until the mushrooms are browned, about 11 to 12 minutes. Sprinkle the mushrooms with the flour and cook, stirring, 1 minute. Pour in the broth, bring to a boil, and cook until the liquid is slightly reduced and thickened, 4 to 5 minutes. Add the heavy cream and cook 1 minute. Remove from the heat and stir in the sour cream, 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper. Transfer to a bowl.
4. Wipe out the skillet. Season the steak with the remaining 3/4 teaspoon salt and 1/4 teaspoon pepper. Heat the oil in the skillet over medium-high heat. Add the steak and cook, turning once, until and instant read thermometer inserted horizontally into the thickest part registers 130F for medium-rare. Transfer the steak to a cutting board and let rest 5 minutes before thinly slicing against the grain.
5. While the steak rests toss the noodles with the mushroom sauce. Divide the noodles among 4 plates and top each with the steak. Top the steak with the onion mixture and serve immediately.