Using the tenderizing side of a meat mallet, give the eye of round steaks a good beating. Hit them about 7-10 times; you don't really want to thin them out or break through the meat.
Slice the steaks into strips, then dice them into bite sized pieces. Start boiling water for your noodles, if you're serving with noodles.
Heat the vegetable oil in a large high sided skillet or large enamel cast iron pot over medium-high heat. Put in half of the meat, so that they aren't crowded and will sear. After a minutes, toss the meat in the pan so each side will brown. Remove the meat to a plate and repeat with remaining steak. Season meat with salt and pepper.
Add the carrots, onion, and mushrooms to the pan over medium. Season with a little salt and pepper and saute your veggies until the onion gets a little browned. Add garlic and cook until fragrant, about 30 seconds.
Add butter to the pan, let melt, then add flour and stir for a minute to cook out the raw flour. Splash in ¼ cup beef broth, stir then whisk in the rest to create gravy.
Temper the sour cream with small amounts of the sauce before adding to pan. Stir in mustard, then season with salt and pepper to taste.
Add meat back into pan and serve with buttered wide egg noodles and parsley.