For the pâté, whisk eggs and fry briefly in a pan. Mix well the minced meat and the finely chopped vegetables and season with salt and pepper.
Roll out the puff pastry thinly and cut out two large round plates, one for the bottom and one for the top.
Place the filling on the bottom, moisten the edges of the pastry and place the lid on top. Press the edges firmly and prick the top several times with a fork to allow the steam to escape during baking.
Brush the surface with the beaten egg and bake the pie for a good 40 minutes in a preheated oven at 170 degrees.