Wipe the chestnut mushrooms clean with kitchen paper and slice them. Crack the free-range eggs into a bowl and add the whole milk. Whisk this with a whisk until smooth and season with salt and freshly ground black pepper.
Melt the butter in a frying pan over medium heat. Add the sliced mushrooms and fry them for about six minutes until golden brown and cooked through. Stir occasionally.
Turn the heat down to low and pour the egg mixture over the mushrooms. Wait briefly until the edges set and push the egg with a spatula gently from the edge to the centre of the pan. Repeat until the scrambled eggs are mostly cooked but still glossy. Then remove the pan from the heat immediately.
Stir the finely chopped fresh chives through the soft egg. Divide the scrambled eggs to the plates immediately to prevent it from cooking further from the residual heat and becoming dry.