Preheat the oven to 350°F.
Season the beef with salt and pepper, then coat it lightly in flour.
Heat butter and oil in a large Dutch oven over medium-high heat.
Brown the beef in batches, making sure not to overcrowd the pan. Remove and set aside.
In the same pot, add onions and garlic. Cook for 2–3 minutes until softened.
Stir in the tomato paste and cook for another minute to deepen the flavor.
Return the beef to the pot.
Add carrots, potatoes, mushrooms, thyme, and rosemary.
Pour in the beef broth and bring to a gentle boil.
Cover and transfer to the oven.
Cook for about 2 hours, until the beef is tender and the vegetables are soft.
Taste and adjust seasoning before serving.