Pat the beef dry and season all over with salt and pepper.
In a heavy skillet over high heat, heat the oil. Add the beef in batches (don’t overcrowd the pan) and brown on all sides — about 3–4 minutes per batch. Transfer browned beef to a slow cooker as done.
In the same skillet, melt the butter over medium heat. Add the sliced onion and minced garlic and sauté for about 3 minutes until softened.
Sprinkle the flour over the onion-garlic mixture and stir for 1 minute, then add the Dijon mustard and stir again — it will look thick and sticky.
Gradually pour in half of the beef stock while stirring to dissolve the flour, forming a smooth gravy. Then add the remaining stock, scrape up any browned bits from the pan, and bring to a simmer briefly.
Pour this mixture over the browned beef in the slow cooker. Cover and cook on LOW for about 8 hours (or on HIGH for ~5 hours) until the beef is fall-apart tender.
While the beef cooks, prepare the mushrooms: in a skillet, melt butter over high heat. Add sliced mushrooms and garlic, sauté until mushrooms are golden and have released their moisture (about 5–7 minutes). Season with salt and pepper.
When the beef is done, turn off the slow cooker. Stir in the sour cream gently. Then fold in the cooked mushrooms. Gently warm if needed — do not boil after sour cream or the sauce may split.
Serve over egg noodles, pasta, mashed potato, or rice. Garnish with chopped chives or parsley.