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1
Combine onion, carrot, and celery in a bowl. Add garlic, rosemary, salt, black pepper, and cayenne pepper. Drizzle in olive oil. Mix stuffing well.
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2
Trim away the wing tips of the chicken. Season the whole surface and cavity generously with salt. Spoon the stuffing into the cavity, reserving any extra for the sauce later. Tie the legs together with string. Refrigerate until ready to wrap.
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3
Place dough onto a well floured surface. Sprinkle more flour on top. Roll out into a large circle about 1/8 inch thick. Place chicken, breast-side down, in the center of the dough. Wrap dough over the chicken. Cut off sections of excess dough and patch them over areas with less coverage.
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4
Beat egg and water together and brush some egg wash over the redistributed dough. Press the rest of dough into the new seams. Continue folding dough over the chicken, brushing with egg wash and patching pieces together as needed, until chicken is fully covered. Do not stretch the dough too thin.
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5
Flip chicken over onto a baking sheet lined with a silicone mat. Press on the inside of both legs to enhance their shape. Refrigerate until crust is cold, about 30 minutes.
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6
Preheat the oven to 400 degrees F (200 degrees C). Brush the remaining egg wash all over the chicken. Sprinkle fleur de sel on top.
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7
Bake in the preheated oven until crust is beautifully browned, about 1 hour and 10 minutes. Transfer chicken to a pie dish and let rest while preparing the sauce, 30 minutes to 1 hour.
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8
Cut a small hole into the back of the chicken and spoon out 1/2 cup of the stuffing. Place in a saucepan with 1 tablespoon butter. Add a pinch of salt. Cook and stir over medium-high heat until vegetables are tender, about 3 minutes. Add wine and the pan drippings. Cook, stirring occasionally, until reduced by half, about 4 minutes. Add cold butter. Turn off heat and stir sauce vigorously until butter melts.
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9
Slice chicken and serve with the sauce.
