For the shortcrust pastry, put the flour and salt in a large bowl and add the cubes of butter and lard.
Use your fingertips to rub the butter into the flour until you have a mixture that resembles coarse breadcrumbs with no large lumps of butter remaining. Try to work quickly so that it does not become greasy.
Measure out four tablespoons of cold water. Using a knife, stir in just enough of the cold water to bind the dough together.
Wrap the dough in cling film and chill in the fridge for 10-15 minutes before using.
Alternatively using a food processor, put the flour, butter and salt in the food processor and pulse until the fat is rubbed into the flour. With the motor running, gradually add the water through the funnel until the dough comes together. Only add enough water to bind it and then stop. Wrap the dough in cling film as before and chill for 10-15 minutes before using.
For the pie, preheat the oven to 220C/425F/Gas 7.
Place the chicken in a large, deep pan with half of the onions and the bay leaf. Season with a little salt and pepper and cover with water. Bring to the boil, then reduce the heat and simmer for 45 minutes, or until the chicken is cooked through.
Remove the chicken from the pan and set aside to cool. Return the pan to the hob and cook the liquid on a rapid boil for a further 30 minutes, or until it has reduced in volume by half. Meanwhile, remove the meat from the chicken legs, thighs and carcass, then place in a 20x30cm/8x12in pie dish.
Heat the olive oil and butter in a large frying pan. Add the remaining onions and cook over a low-medium heat, without browning, for about five minutes, or until softened, then add the flour. Turn up the heat and then add the mushrooms and cook for a further 3-4 minutes.
Strain the reduced stock, then add it to the mushrooms and cook for one minute. Pour in the wine and cream, then bring to the boil, reduce the heat and simmer for five minutes, or until nicely thickened. Season, then pour it over the chicken and sprinkle tarragon leaves over the top.
Roll out a rectangle of pastry 2cm/¾in wider than the pie dish. Brush the edges of the dish with beaten egg and lay the pastry on top, crimping the edges with a fork or your fingertips. Trim away any excess pastry and brush the top of the pie with the remaining egg. Place on a baking tray and bake for 30-35 minutes, or until the pastry is crisp and golden-brown.
For the carrots and peas, heat a frying pan until hot. Add the butter, sugar and five tablespoons of water and cook until the volume of liquid has reduced slightly. Add the carrots and cook for a couple of minutes, then add the peas and continue to cook until they are tender and the liquid has reduced to a glaze.
To serve, remove the pie from the oven and allow to cool slightly before serving with the carrots and peas.