The night before cooking, season the chicken liberally with sea salt inside and out. Leave uncovered overnight in the refrigerator.
In a large bowl, mix the flour with ⅓ cup water and knead into a smooth dough. Cover and chill overnight.
The next day, preheat oven to 350°F. Bring chicken to room temperature 30 minutes before you intend to cook. Thoroughly pat the skin dry with paper towel. Heat oil in a large Dutch oven over medium-high. Add the chicken on its side so that the leg and thigh is in contact with the pot. Brown on its side for 10 minutes, then flip and repeat on the other side. Flip onto its back and brown for 5 minutes, then, finally, brown the breasts for 5 minutes. Set aside.
On a floured bench, roll the dough into a log long enough to wrap around the circumference of the Dutch oven.
Place half the hay into the Dutch oven, then place the chicken on top. Brush the chicken all over with the butter, then cover with the remaining hay. Seal the lid with the rolled dough, then brush with the egg and bake for 45 minutes. Remove from the oven and rest, still sealed, for at least 30 minutes.
Once rested, break into the dough and remove the lid. Carve the chicken into eight pieces, then serve on a warmed platter.
Note: Begin one day in advance
Note: if preparing this as part of the full Thanksgiving menu, bake the chicken before beginning the confit sole. Reduce oven heat to the confit temperature while the cooked chicken rests.