Bring the water to the boil in a saucepan with the vinegar and a pinch of salt.
Carefully crack the eggs into the boiling water, one at a time, and poach for 2-3 minutes, until the white sets but the yolk stays soft.
Lift out with a slotted spoon, plunge into cold water, then drain on a sieve, and trim off any ragged edges.
For the sauce, make a light roux from the butter and the flour, stirring for 1-2 minutes, add the broth, mix smooth, and add the finely chopped mushrooms.
Salt it, and cook over low heat for 8-10 minutes, until the mushrooms are tender and the sauce has thickened.
Pass through a fine sieve, or purée with an immersion blender, if you'd like a thicker but more textured sauce.
Return to the heat, stir in the egg yolk, the 1 tbsp butter and the cream, and heat gently for 1-2 more minutes, stirring constantly, but don't let it boil, or the yolk will curdle.
Add the cognac and a pinch of cayenne pepper.
Dip the cooled, poached eggs one by one, quickly, into the still-hot sauce, then lift out with a spoon onto a flat plate or a parchment-lined tray.
Chill for at least 30 minutes, until the sauce layer sets on the eggs.
Coat the sauce-covered eggs in breadcrumbs, then in beaten egg, then in breadcrumbs again.
Fry quickly in plenty of hot (about 180°C) oil or lard for 1-2 minutes, until golden brown, being careful that the inside of the egg stays soft.
Drain on paper towel, and serve immediately, hot.
Based on: Malatinszky Fanny szakácskönyve (a classic Hungarian cookbook), modernised. Calories and protein are estimated.