Whisk the eggs with the ice-cold water, pass through a sieve, and keep in the fridge.

Sift the flour and salt into a bowl, add the cold butter, and rub it in until the mix is sandy but still shows small nuggets of butter. Those irregular pieces are what make the pastry flaky, so do not rub it to a uniform powder.

Mix in the chilled egg with a fork until it just starts to come together and is still crumbly. Do not overwork it.

Tip onto a surface and press together without kneading. Wrap and chill at least 1 hour, ideally overnight.

Break down both chickens. Put 4 legs and 2 breasts on one tray for the pie and reserve the other 2 breasts for another use. Put the bones, skin, and wings on a second tray.

Rub the legs and breasts with oil and salt and roast at 230°C (445°F) for about 20 minutes, pulling the breasts at 69°C (156°F) and the legs at 74°C (165°F). Save every drop of the juices and move the skins onto the bone tray.

Roast the bone tray until deep amber brown. Pour off and reserve the rendered fat.

Deglaze the bone tray with the white wine, scraping up all the fond.

In a large pot, sweat the carrot, leeks, celery, and garlic in a little rendered fat with no salt. Add the shiitake, bay, and peppercorns, then the roasted bones, the deglazing liquid, and the saved juices. Cover with just enough water.

Bring to a steady simmer, skim, and reduce by half to a concentrated, full-bodied stock. Do not salt it. Strain and measure out 600 ml for the velouté.

Pick the roasted meat, tearing the breast roughly and running a knife through it so there are no stringy pieces. Keep it chunky. Add the picked bones to the simmering stock.

Brown the mushrooms hard in the oil or rendered fat with no salt, stirring for even color. Add the leek, crushed garlic, thyme, and salt and cook until soft. Add the soy and a splash of stock and simmer until the liquid has evaporated.

Fold the mushroom and leek base through the picked chicken.

Melt the butter and rendered fat, add the flour, and cook to a light blonde roux.

Ladle in the 600 ml reduced stock bit by bit, whisking smooth after each addition. Off the heat, whisk in the cream, Dijon, nutritional yeast, and salt.

Fold the velouté through the chicken and mushrooms, then stir in the chopped tarragon.

Cool the filling completely until thick and set. Never fill the pastry warm. This is best made a day ahead.

Let the pastry sit until about 13°C (55°F), then roll it to 4 mm thick. Butter a 23 cm (9 in) springform generously. Cut a base larger than the tin and a lid slightly larger than the top.

Line the tin with the base, pushing it into the walls with no trapped air, leaving a 1.5 cm overhang. Chill 10 minutes, and chill the lid too.

Spoon in the cold filling, leaving 2 cm clear at the top. Lay on the lid and press the edges together to seal.

Whisk the egg yolks with the water for the wash. Brush the lid thinly, crimp the edges, and cut a central steam vent. Chill 10 minutes, then egg-wash once more.

Bake at 200°C (390°F) fan for 40 to 45 minutes, rotating halfway, until deep golden. Rest 15 to 20 minutes, then release the springform and cut a big slice.