Preheat the oven to 400 degrees . In a large skillet, melt 2 tablespoons butter over medium-high heat. Add the onion and thyme and cook for 3 minutes. Add the mushrooms and cook, stirring, until golden, about 3 minutes. Stir in the flour for 1 minute, then add the wine. Simmer for 5 minutes, then add the broth and bring to a boil. Cook until thickened, about 15 minutes; season with salt and pepper. Cover to keep warm.
In another large skillet, melt the remaining 1 1/2 tablespoons butter. Season the steaks with salt and pepper and cook for 4 minutes on each side for medium-rare. Transfer to a cutting board.
Spread the brie on the cut sides of the baguette; place on a baking sheet. Toast in the oven to melt the cheese. Slice each half crosswise into quarters.
Fill a deep skillet with water; bring to a slow simmer. Crack 1 egg into a cup. Stir the water, creating a swirl; slide the egg into the water; cook, stirring, until the white is firm, 1 to 1 1/2 minutes. Using a slotted spoon, transfer to a damp dish towel. Season with salt and pepper. Repeat with the remaining eggs.
Thinly slice the steak and divide among 4 plates. Top with the mushroom sauce. Serve with the eggs and brie toasts.