For the dough, heat the milk to room temperature and then add the honey and yeast. Wait for the yeast to activate (you will see some froth), then add the cooled melted butter, egg yolks and salt. Whisk until smooth and all the ingredients are incorporated.
Add the flour gradually, first whisking it in, then working it with your hands. You will have a very soft and slightly sticky dough. Don’t be scared of it, you will see later that it is very forgiving. Transfer to a clean bowl, cover with cling film and place somewhere warm to rise for a couple of hours, or until doubled in size.
For the filling, put two frying pans on the hob, one larger with a heavy base for the mushrooms and one smaller for your shallots, garlic, cinnamon and apples.
Heat a knob of butter and a little oil in each pan. Put the large pan over a high heat and the smaller one over a medium-low heat. Fry the mushrooms in batches in the large pan, taking care not to overcrowd the pan, and seasoning each batch with a little salt. Add more butter and oil if they start to stick.
Meanwhile, in the smaller pan, gently fry the onions and cover with a circle of baking parchment to help them soften faster. Cook over a low heat for about 5 minutes, then add the apple, garlic and cinnamon and cook for another couple of minutes. Season with salt and add to the mushrooms.
Crumble the chestnuts into the filling, mix it well and taste. Season with a touch more salt, if required. Now grind in quite a lot of black pepper. (We are using it as spice here, not just as seasoning. The filling should have a little bit of heat.)
Preheat the oven to 180F/160C Fan/Gas 4.
When the dough has doubled in size, divide it into 100g/3½oz pieces (weighing them makes sure they are evenly sized). You should have approximately 8 balls of dough.
On a well-floured work surface, roll each piece of dough into a 15cm/6in circle. Then divide the filling equally between each circle (about 3–4 tablespoons each). It might look like too much filling, but the dough is forgiving and malleable so don’t worry.
Bring the edges of the dough circles together and pinch really well to seal in the filling. Place on a baking tray seam-side down. They will merge together as they rise, but that’s fine.
Cover the buns with a clean, damp tea towel and leave to rise again for 10 minutes.
Brush the buns with egg yolk and bake for 15–20 minutes, or until golden and cooked through. Serve hot or cold.