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1
In a large frying pan over medium-high heat, melt the 2 Tbs. butter until foaming. Then, stir in the mushrooms and a generous pinch of salt. Cook until the mushrooms have given off their liquid and it has mostly cooked off (about 6 minutes).
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2
Stir in the thyme and the shallot (add a tiny pat of butter if the pan has gotten too dry) and cook until the shallot begins to soften, another minute or two, then stir in the sherry. Let the sherry cook off, then add the broth. Bring to a boil, then turn down slightly to a simmer, and cook until the broth has been reduced to about 1/4 cup. Then, remove from the heat and set to the side.
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3
Bring a pan with a couple of inches of water to a boil and add the splash of vinegar. Take off the heat and once the bubbles die down, immediately slide in the two eggs from their cups (gently and on opposite sides of the pan - you may need to use a spatula to keep them apart), and allow to sit for 4-6 minutes, at this point the whites should be set but the yolk still runny. Then remove the eggs from the water with a slotted spoon and set on a paper towel to dry slightly.
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4
While the eggs are sitting in the poaching water toast your brioche slices until golden brown. Put each slice on a plate. Brush with the melted butter and sprinkle each piece with chives and chervil.
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5
As soon as the eggs are about ready, bring your pan with the mushrooms back up to a simmer, then swirl in the creme fraiche. Spoon the 'shrooms and sauce over the toasts and top each with a poached egg. Sprinkle with some sea salt and freshly ground pepper and enjoy. (And, if you happen to accompany your toast with some crisp tender asparagus, I can pretty much promise that you will not be sad about it.)
