Preheat the oven to 180ºC, fan 160ºC, gas 4. In a large saucepan, heat the oil and cook the onion for 5 minutes, until softened. Add the garlic, carrots and bacon and cook for a further 5 minutes.
Meanwhile, make up the parsley sauce to pack instructions using the milk. Stir it into the pan with the bacon mix, and add the broccoli and peas. Season with freshly ground black pepper and transfer the mixture to a 1 or 1.5 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, and 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.