Heat a medium sized heavy pan over moderate heat.
Add the oil and bacon. Cook until fragrant and gently sizzling.
Increase the heat, add half the butter to the bacon and allow to foam. Throw in the mushrooms, leave to cook without stirring for a couple of moments, then toss around using a fork or such. The mushrooms will give up some liquid depending on their water content. Let this liquid completely evaporate. As the mushrooms take on a little colour, season lightly with salt and pepper. As soon as they are tender tip them out into a small colander set over a bowl and keep them warm.
Break the eggs into a bowl; whisk them vigorously using a fork until the yolks and whites are just blended.
Place a heavy pan over moderate heat and smear the base and sides with some of the oil from the bacon and mushrooms.
Pour in the eggs, lower the heat, and, using a heat proof rubber spatula, stir the eggs from the bottom of the pan slowly but constantly. As soon as you feel the eggs thickening on the base of the pan remove from the heat and continue stirring.
Return the eggs to the heat and repeat the process. The idea is to cook the eggs slowly so that they don’t become dry and rubbery but soft and silky.
When you are satisfied that the eggs are almost done (better under than over) remove from the heat and gently stir in the cream, remaining chilled butter and then the bacon and mushrooms, check the seasoning.
Serve on thickly cut warm toast with the chives sprinkled over.