Place chicken on cutting board breast-side down. Use kitchen shears to cut down the ribs on one side of the backbone and then the other and remove. Open the bird like a book and remove the keel bone separating the breast halves by slicing through the thin membrane covering it (I like a thin paring knife for this), then work two fingers under the bone and lever it out. Flip chicken breast-side up and flatten it, pulling the legs toward you, so that the bird looks like a big, meaty butterfly. Now wash those chicken-y hands!
Place the chicken on an oven-safe wire rack set in a rimmed half sheet pan. Rub on all sides with the salt, then flip skin side-up. Refrigerate, uncovered, overnight.
The next day, adjust one oven rack to the top position and place pizza stone(s) on it. Adjust the second rack 6 inches below the top rack then crank the oven to 500°F. Give the oven and stone(s) at least an hour to come up to temperature. Your patience will be rewarded.
While the oven heats, remove the chicken from the fridge and let come to room temperature on the counter.
When the oven is ready, combine the butter, Aleppo pepper and za'atar in a heavy metal measuring cup or small saucepan and cook over low heat until the butter is red and fragrant, about 5 minutes.
Flip the chicken skin-side down, brush with half the butter mixture, and then transfer, still on the wire rack and baking sheet, to the oven rack below the pizza stones and roast for 10 minutes.
Carefully (but quickly) slide out rack, flip the bird skin-side up and brush the skin with the remaining butter mixture. Insert a remote probe thermometer into the breast and set the alarm to 160°F. Now close the door!
Roast until the alarm beeps or a separate digital instant read thermometer registers 160°F. Remove the chicken from the oven and transfer to a cutting board. Let rest for 10 to 15 minutes before carving and serving.