1. Heat a large pot of water.
String and wash the green beans.
Add them to the boiling water.
2. Add salt and let cook for 30 minutes.
Meanwhile, prepare the white sauce:
Heat 50 cl of milk over medium heat. Mix 4 tablespoons of flour and 3 tablespoons of water in a large bowl.
3. Stir with a wooden spoon until there are no more lumps.
Add a small ladle of hot milk to the bowl, mix, then pour the contents of the bowl into the pot.
Immediately reduce the heat and stir constantly for 3 to 4 minutes until the sauce is thick. Season with salt and pepper.
4. Set aside.
Preheat the oven to 220°C (approximately 425°F).
Drain the green beans when they are well cooked. Put them in the blender to puree them.
Add the white sauce to this puree and mix.
Separate 2 eggs, keeping the whites for another use.
5. Add the yolks to the green bean puree and mix well.
6. Butter a 20cm x 18cm gratin dish with a teaspoon of butter.
Pour the mixture into the dish.
7. Sprinkle with 4 teaspoons of breadcrumbs.
Cut the equivalent of 2 teaspoons of butter into small pieces.
8. Spread them on top of the dish.
Grill for 20 minutes in the hot oven.