Heat the oil in a large stockpot over medium-high heat.
Season the meat generously with salt and pepper. Add the meat to the hot oil and cook until browned on all sides, about 5–7 minutes. You may need to brown the beef in batches to get a good sear. If so, drain any accumulated grease between batches.
Add the onion, bell pepper and mushrooms to the pot and sauté until tender, about 3-5 minutes.
Add the garlic and sauté for 30 seconds.
Add the broth to the pot and scrape up any browned bits from the bottom of the pan.
Stir in the brown gravy mix until dissolved. Add the butter and pepperoncini peppers.
Bring the liquid to a boil, then cover the pot and reduce the heat to low. Simmer for 1 hour and 15 minutes.
Remove the lid and simmer for an additional 15 minutes, allowing the gravy to thicken.
Remove the pepperoncini peppers before serving. Discard them, or remove the stems, chop the peppers, and stir them back into the gravy.
Taste and season with additional salt and pepper, if needed. Serve over hot egg noodles, white rice, or mashed potatoes.
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Notes
Don't rush the browning step. Searing the beef develops deep, rich flavor that forms the foundation of the gravy. Brown the meat in batches if necessary so it sears instead of steams.
Chuck roast makes an excellent substitute. If your grocery store doesn't carry stew meat, cut a boneless chuck roast into 1½-inch cubes. The extra marbling makes for especially tender beef tips.
Pepperoncini peppers won't make the dish spicy. They add a mild tang that balances the rich gravy while helping tenderize the beef during the long simmer.
The gravy will continue to thicken as it rests. If it seems a little thin when you remove the pot from the heat, let it sit for 10 to 15 minutes before serving.
If the beef is still tough, keep cooking. Tough stew meat simply needs more time for the connective tissue to break down. Continue simmering in 15-minute increments until the beef is fork tender.
Store side dishes separately. Egg noodles, rice, and mashed potatoes will absorb the gravy during storage, so keep them separate from the beef tips if possible.
Leftovers are even better the next day. The flavors continue to develop overnight, making this an excellent make-ahead meal.
For a richer gravy, stir in 1 to 2 tablespoons of Worcestershire sauce or replace up to ½ cup of the beef broth with a dry red wine such as Cabernet Sauvignon or Merlot.