Season the beef. Pat the beef dry with paper towels. Season with salt and pepper.
Sear the beef. Heat 1 tablespoon olive oil in a large skillet or dutch oven over medium-high heat. Sear the beef in batches for 2-3 minutes per side, until browned. Transfer to a plate.
Brown the mushrooms. Add the remaining tablespoon of olive oil to the skillet. Add mushrooms in an even layer. Cook 3-4 minutes without stirring, then stir and cook 2-3 more minutes.
Cook onion + garlic. Add onion and cook 3-4 minutes, until softened. Add garlic and cook 30 seconds.
Build the gravy. Pour in ½ cup beef broth and scrape up browned bits. Stir in Worcestershire, dijon mustard, thyme, and remaining broth.
Simmer. Add the seared beef (and juices) back into the skillet. Bring to a gentle simmer, reduce heat to low, cover, and simmer 25-35 minutes for sirloin tips, or 60-90 minutes for stew meat, until tender.
Thicken the gravy.
Roux: Melt butter in a small saucepan. Whisk in flour and cook 1-2 minutes. Whisk in 1 cup hot broth from the skillet until smooth, then stir it back into the skillet. Simmer 2-3 minutes.
Slurry: Mix cornstarch with cold water until smooth. Stir into the simmering gravy and cook 1-2 minutes.
Finish and serve. Taste and adjust salt/pepper. Optional: stir in heavy cream. Serve hot over mashed potatoes, egg noodles, or rice.