Poulet Au Pain

3 cups all-purpose flour, plus more as needed
1 1/2 teaspoons kosher salt, plus more as needed
1/2 teaspoon sugar 
12 tablespoons (1 1/2 sticks) unsalted butter, cut into pieces
1 tablespoon olive oil
2 peeled and chopped celery stalks
1 peeled and chopped medium carrot
1  chopped medium onion
1 tablespoon chopped fresh rosemary
1 3- to 3 1/2-pound chicken, giblets and excess fat and skin removed, and patted dry
Freshly ground black pepper
1 large egg, beaten

Make the bread dough: combine the flour, salt, and sugar in the bowl of a stand mixer fitted with the paddle attachment. Add the butter and mix with the paddle, stopping the mixer occasionally to break up bigger chunks of butter with your hand. Add 1/2 cup of water and continue mixing until the dough begins to come together. At this point, turn off the mixer and switch to the dough hook (scraping all the dough off the paddle first, of course). Knead the dough into the hook until it comes together in one mass, 1 to 2 minutes. Turn the dough out onto a lightly floured surface and continue to knead it by hand, pushing it away from you with the heel of your hand, folding it over, giving it a quarter turn, and pushing it away again until it feels nice and elastic. If the dough is very sticky, add a little more flour to it as you knead. Wrap the dough in plastic wrap and refrigerate it for at least 30 minutes.   

Make the stuffing: heat the olive oil in a medium skillet over medium heat. Add the celery, carrots, onions, rosemary, and a pinch of salt and cook, stirring occasionally, until the vegetables are just tender but not colored, 8 minutes. Let cool.

Prepare the chicken: clip the chicken wings off at the body and save them for making stock, or discard. Season the chicken liberally inside and out with salt and pepper. Stuff the bird with the cooled vegetables and tie the legs together with kitchen twine.

Pre-heat the oven to 400° F. On a very lightly floured surface, roll the dough out to 1/8-inch thick; this will take some muscle. If the dough is very stubborn, let it rest for a few minutes before trying to roll it some more. Put the chicken on the dough breast-side-down and wrap the dough up and around the bird, encompassing it completely and overlapping the dough. (If there is a lot of overlap, trim the dough.) Pinch the seams together to keep them closed. Turn the bird over and put the bird seam-side-down on a baking sheet.

Brush the dough all over with the beaten egg and sprinkle it lightly all over with salt. Bake it until the chicken is cooked through and the bread is a lovely golden brown, 1 to 1 1/2 hours, depending on the size of the chicken (an instant-read thermometer inserted through the crust into the breast should read 170° F.) Let the chicken cool for at least an hour, preferably 2, before tearing it apart and serving.

Note: you can refrigerate the dough-wrapped chicken, covered in plastic wrap, for a day before cooking it.