Heat olive oil and butter together in a skillet over low heat until smooth.
Whisk flour, salt, and pepper together in a shallow bowl. Coat beef strips in flour mixture, gently shaking off any excess flour.
Increase heat under skillet to medium. Add beef to skillet and cook and stir until browned, 1 to 2 minutes per side. Mix mushrooms into beef, adding more oil if pan is dry; cook and stir until mushrooms are lightly browned, about 5 minutes.
Pour wine into the skillet and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon; continue cooking until liquid is reduced by half, 5 to 10 minutes. Add beef broth and return to a boil.
Cover skillet and cook until beef is tender, 20 to 25 minutes.
Bring a large pot of lightly salted water to a boil. Cook egg noodles in the boiling water, stirring occasionally until cooked through but firm to the bite, about 5 minutes. Drain.
Mix onions into beef mixture; cook and stir until onions are warmed, 3 to 5 minutes. Remove skillet from heat and gently stir sour cream into beef mixture; serve over noodles.