Cook the broad beans in salted water, drain, and peel off the skins (you should have about 150 g peeled beans). Meanwhile, shred or cut the cooked chicken into bite-sized pieces.
Melt 1 tablespoon of butter in a pan and sauté the finely chopped leek. Add the potatoes cut into small cubes and the green peas. Cook for a few minutes, until the vegetables are al dente. Add the cooked chicken, broad beans, and thyme, stir to combine, and remove from the heat.
Preheat the oven to 190°C.
In a saucepan, melt the butter, add the flour, and stir briefly. Gradually pour in the broth and milk, whisking constantly. Season with salt and cook over low heat until the sauce thickens slightly.
Combine the sauce with the chicken and vegetables, then transfer the mixture to an ovenproof dish. Trim the puff pastry to fit the dish and place it over the top. Cut a small slit in the center and brush the pastry with beaten egg.
Bake for about 30 minutes, until the pastry is golden and crisp.