In a stand mixer fitted with the paddle attachment, combine the flour, sugar, and 1-1/2 tsp. salt. Add the butter and mix on low speed, stopping the mixer occasionally to break up bigger chunks of butter with your fingers, until the mixture resembles coarse meal with some pea-size pieces. (If you don’t have a stand mixer, use a pastry cutter or your fingers to work the butter into the dry ingredients.) Add 1/2 cup water and continue mixing (or stir with a fork) until the dough begins to come together, about 1 minute more. Turn the mixer off, scrape the dough off the paddle, and switch to the dough hook. Knead the dough on medium-low speed (or by hand) until it comes together in a mass, 1 to 2 minutes (longer by hand).
Turn the dough out onto a lightly floured surface and continue to knead it briefly until it feels smooth and elastic. If the dough is very sticky, add a little more flour.
Wrap the dough in plastic and refrigerate it for at least 30 minutes and up to 2 hours.
Position a rack in the lower third of the oven and heat the oven to 400°F. Line a large rimmed baking sheet with parchment.
Meanwhile, heat the olive oil in a heavy-duty 10-inch skillet over medium heat. Add the celery, carrot, onion, rosemary, and a pinch of salt and cook, stirring often, until the vegetables are just tender but not colored, about 8 minutes. Let cool.
Remove the giblets from the chicken and discard or save for another use. Pat the chicken dry and trim away any excess fat. Cut off the first two joints of the wings and discard or save for another use. Season the chicken liberally inside and out with salt and pepper. Put the vegetables inside the cavity and tie the legs together with kitchen string.
On a very lightly floured surface, roll the dough out to a 1/8-inch-thick, 18x13-inch rectangle. (If the dough shrinks back, let it rest for a few minutes at room temperature, then roll again.) Put the chicken on the dough breast side down and wrap the dough around the bird, overlapping the edges. If there is a lot of overlap, trim the dough. Pinch the seams together to seal them. Turn the bird over and put it seam side down on the prepared baking sheet. Gently mold the dough over the contours of the chicken to delineate its shape.
Brush the tops and sides of the dough with the egg wash and sprinkle with salt.
Bake the chicken until the pastry is golden-brown (if it’s browning too much, tent it loosely with foil) and an instant-read thermometer inserted through the crust and into the breast registers 160°F, 1 to 1-1/2 hours. Let cool for at least 1 hour (the chicken will still be quite warm after 1 hour) and up to 3 hours before serving. Serve warm or at room temperature.
To carve the chicken, crack into the crust with a knife and carve away pieces of the chicken, starting with the breast. Serve the chicken with some of the crust and the vegetable stuffing.
You can wrap the chicken in dough up to 1 day ahead; wrap it in parchment and refrigerate. Brush with egg wash just before baking.
You can let the fully baked chicken rest at room temperature for up to 3 hours before serving (though the meat and vegetables will be closer to room temperature after resting this long).