1. Heat the oven to 350 degrees F.
2. In a large Dutch oven, melt 2 tablespoons of the butter over medium heat until foaming. Season the flank steak on both sides with salt and pepper. Brown the steak in the butter, one piece at a time, about 2 to 3 minutes per side.
3. As each piece is browned, remove it to a clean plate. When all of the steak is browned, add the remaining tablespoon of butter to the pot, then the onion, garlic, and mushrooms. Season vegetables generously with salt and pepper and cook until softened, about 5 minutes, stirring occasionally. Add the brown sugar and stir until it melts.
4. Add wine to pot and bring it to simmer. Cook, stirring occasionally, until all of the wine evaporates, about 5 minutes. Add the tomatoes and tarragon. Bring the mixture back up to a simmer and put the steak back into the pan. (The liquid should come halfway up the sides of the beef.) Cover the pot and transfer it to the oven to cook for 12 minutes (See Note).
5. Remove the pan from the oven. Transfer the meat to a cutting board and let it rest for 5 minutes. (If the liquid looks too soupy, remove beef from pot and reduce the liquid over medium-high heat until it has thickened.) Slice the steak very thin and serve with sauce and vegetables.