Preheat oven to 300 degrees.
Pour flour in a bowl, and add salt and pepper. Whisk to combine.
Dredge the steaks in flour on both sides. Set aside.
In a large oven-safe skillet over medium-high heat, add olive oil and brown steaks, in batches if necessary, on both sides; about 2 minutes per side. Set aside and keep warm.
Meanwhile, in a saucepan, reduce beef stock by half.
Season mushrooms with salt and pepper to taste.
Saute sliced mushrooms in the same pan as the meat, in the leftover olive oil, until very brown, about 5 minutes.
Add butter, minced onion, carrot, thyme and garlic. Cook until vegetables are caramelized, about 5 minutes.
Add balsamic vinegar and reduce for 1 to 2 minutes.
Add broth, and slide the meat back into the sauce.
Cover and bake until tender, about 1 hour.
Serve hot.