In a large nonstick skillet, heat 2 tablespoons olive oil over medium heat. Add the onion and allspice and cook, stirring occasionally, until the onion is softened, about 8 minutes. Stir in the broth, gingersnaps and vinegar and cook until thickened, about 6 minutes. Season with salt and pepper; transfer to a medium bowl, cover and set aside. Wipe out the skillet and set aside.
Bring a large pot of salted water to a boil. Add the noodles and cook until al dente, about 8 minutes. Drain, return to the pot and toss with 3 tablespoons olive oil. Set aside.
Meanwhile, season the steak with salt and pepper. In the reserved skillet, heat the remaining 2 tablespoons olive oil over medium-high heat. Add the steak and cook for 2 to 3 minutes on each side for medium-rare. Transfer to a cutting board, tent with foil and let rest for 5 minutes. Cut crosswise into 1/4-inch-thick slices.
Pour off the fat from the skillet, add the reserved sauce, scraping up the browned bits, and cook over medium heat until warmed through. Divide the noodles among 4 shallow bowls and top with the steak and sauerbraten sauce.