Place the sour cream (or Greek yogurt) in a medium bowl. Set aside to let it come to room temperature while you prepare the rest of the recipe.
Heat the olive oil in a large skillet over medium-high heat. Once hot, add the steak strips to the pan in a single layer. (Cook in two batches if needed.) Sprinkle steak with salt and pepper. Sear on the first side, undisturbed, for 1-2 minutes, then flip and cook on the second side for 1-2 minutes. (You just want to brown the steak, it can finish cooking in the sauce.) Remove steak to a clean plate and loosely cover with foil to keep warm.
Add the butter to the pan. Once melted, add the onion. Cook, stirring often, for 2 minutes. Add the mushrooms and cook, stirring occasionally, for 6-8 minutes, until softened and browned. Season with some salt and pepper.
Add the garlic and flour and cook, stirring, for 30 seconds. Stir in the beef broth, Dijon mustard and dried thyme. Remove 3-4 tablespoons of the hot broth from the pan and stir it into the sour cream. (This tempers the sour cream so that it won't curdle when you later add it to the hot sauce.) Set tempered sour cream aside.
Bring the pan to a simmer and cook for 2-3 minutes, until the sauce thickens, stirring often and scraping up any browned bits from the bottom of the pan. Return the beef to the skillet and cook for about 1 minute.
Remove the pan from the heat. Stir in the tempered sour cream. Taste and season with more salt and pepper if needed.
Serve beef stroganoff over noodles, mashed potatoes or rice, garnished with fresh parsley, or as desired.