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1
Season the chicken pieces with salt and white pepper. Melt half the butter in a wide, heavy casserole over a moderate flame. Add the chicken in a single layer and sear, turning once, until the pieces reach a pale gold — not a deep caramelization. The skin should show colour without taking on any brown. Remove and set aside.
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2
Sprinkle the flour directly into the butter and meat juices remaining in the casserole. Stir continuously for one minute over a low flame, scraping the base, to form a white roux. The paste should smell faintly of cooked flour but carry no colour.
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3
Pour in the dry Vouvray in a steady stream, whisking to dissolve the roux without lumps, then add the chicken stock. Return the chicken pieces and the bouquet garni. Bring to a bare simmer, cover, and braise gently for 30–35 minutes. The meat is done when the juices run clear and the flesh yields readily to the tip of a small knife.
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4
While the chicken braises, melt half the remaining butter in a sauté pan and cook the mushrooms over a lively heat until golden — not grey, not steaming. In a separate small pan, combine the pearl onions with the last of the butter and enough water to barely cover; simmer, uncovered, until the water evaporates and the onions become tender and shiny (glazed à blanc). Reserve both.
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5
This is where the dish lives or dies. Remove the casserole from the heat entirely. Beat the egg yolks with the crème fraîche in a bowl. Ladle two spoonfuls of the hot braising liquid into the egg-cream mixture, stirring constantly to temper it; then pour the tempered liaison back into the casserole, stirring without pause. The sauce will thicken to a velvety, ivory cream. It must not return to a boil — above 82 °C the liaison will scramble and the sauce is lost.
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6
Fold the mushrooms and glazed pearl onions into the sauce. Scatter the freshly chopped tarragon over at the last moment. Taste and adjust the seasoning. The finished sauce should be simultaneously rich and bright, the Vouvray's acidity cutting through the cream.