Season the beef cubes lightly with salt and pepper. Heat 2 tablespoons of the butter in a heavy 6-quart pot over medium heat. As soon as the butter starts to turn brown, add half the beef and raise the heat to high. At first, the beef will give off some liquid, but once that evaporates, the beef will start to brown. Cook, turning the beef cubes on all sides until the pieces are as evenly browned as possible, about 5 or 6 minutes after the water has boiled off. If the pan starts to get too brown at any point, just turn down the heat a little. Scoop the beef into a bowl and brown the rest of the beef the same way using the remaining butter.
Scoop out the second batch of beef, then add the carrots and onions and raise the heat to medium-high. Cook until the onion starts to turn translucent, about 5 minutes. Stir in the flour until it has been worked into the veggies and you can’t see it any more. Pour in the chicken broth, wine, and crushed tomatoes, and toss in the rosemary. Slide the beef back into the pot and bring the liquid to a boil.
Turn down the heat so the liquid is just breaking a gentle simmer. Partially cover the pot and cook 50 minutes. Stir the stew several times while simmering so it cooks evenly and nothing sticks to the bottom.
Stir the potatoes into the stew, cover the pot completely, and cook until the potatoes and beef are tender, stirring occasionally, about another 45 minutes. Add the beans and cook for another 5 minutes until the green beans turn bright green and are cooked through but still have a nice snap to them.