Preheat the oven to 180ºC/fan 160ºC/gas mark 4. Heat the oil in a large saucepan and cook the onion for 5 minutes until softened. Add the carrots and bacon and cook for a further 5 minutes.
Meanwhile, make up the parsley sauce according to the instructions on the packet using the milk. Stir it into the pan with the bacon mix, then add the broccoli and peas. Season with freshly ground black pepper and transfer the mixture to a 1-litre pie dish.
Unroll the puff pastry and cut into 2cm-wide strips. Arrange the strips over the top of the pie filling, trimming any excess pastry, then brush all over with the beaten egg. Bake in the oven for 25-30 minutes, until the pastry is golden and the filling is bubbling.