Place the pizza stone in a gas oven on a rack about 8 inches from the broiler. Preheat the oven on bake at 500 degrees F for 30 minutes. Switch to broil for 10 minutes.
Cut away about 2 inches of the base of each asparagus spear.
With a vegetable peeler, shave the entire asparagus from bottom to top, reversing your grip and rotating as necessary to shave as much as possible. Don't rush it; be deliberate for the greatest precision. You should have about 90 grams (3 ounces) of very thin ribbons.
With the dough on the peel, sprinkle the Parmigiano evenly over the surface and distribute the chunks of Saint Nectaire on top. Arrange the asparagus shavings over the cheese.
With quick, jerking motions, slide the pie onto the stone. Broil for 21/2 minutes under gas. The cheese should be bubbling, the crust only slightly charred.
Using the peel, pull the pizza out of the oven. Close the oven to conserve heat.
Crack the eggs very carefully to keep each yolk whole. Place the eggs around the pizza (one for each slice). Sprinkle the salt evenly over the pie. Return to the oven to broil for 1 minute, until the eggs are set but not hard and the charring is deeper.
Using the peel, transfer the pizza to a tray or serving platter. Slice into wedges (cutting through the egg yolks to allow them to spread slightly). Serve immediately.
For the Dough:
In a medium bowl, thoroughly blend the flour, yeast, and salt. Add the water and, with a wooden spoon or your hands, mix thoroughly. Cover the bowl with plastic wrap or a kitchen towel and allow it to rise at room temperature (about 72 degrees F) for 18 hours or until it has more than doubled. It will take longer in a chilly room and less time in a very warm one.
Flour a work surface and scrape out the dough. Divide it into 4 equal parts and shape them (see Cook's Note): For each portion, start with the right side of the dough and pull it toward the center; then do the same with the left, then the top, then the bottom. (The order doesn't actually matter; what you want is four folds.) Shape each portion into a round and turn seam side down. Mold the dough into a neat circular mound. The mounds should not be sticky; if they are, dust with more flour.
*If quail eggs are unavailable (farmer's markets are often an excellent source), go with the smallest eggs you can find. In the restaurant I dress up the pie more than I do here, with a slice of oil-packed truffle on each yolk. But those truffles in oil are virtually impossible to find retail. If you're feeling flush, go ahead and put a slice of truffle on each yolk or, for that matter, a drizzle of truffle oil (but not both, which would be overwhelming). In any case, the way I present it here-truffle-less-is great. While I'm not picky about the flour--either bread flour or all-purpose is fine--what does concern me is how the dough is handled. Treat it gently so the dough holds its character, its texture. When you get around to shaping the disk for a pie, go easy as you stretch it to allow it to retain a bit of bumpiness (I think of it as blistering), so not all of the gas is smashed out of the fermented dough. I prefer to hold off on shaping the ball until just before topping it. If it's going to sit for a while--more than a couple of minutes--cover it with a damp kitchen towel to prevent it from drying out. I offer you two approaches for shaping. The simpler one, executed completely on the work surface, is slower than the second, where you lift the disk in the air and stretch it by rotating it on your knuckles. Lifting it into the air to shape it is more fun, too. Shaping the Disk (Method 1): Take one ball of the dough and generously flour it, your hands, and the work surface. Then press it down and gently stretch it out to 6 to 8 inches. Very carefully continue the process, massaging it into a roundish disk of 10 to 12 inches, stroking and shaping with the palms of your hands and with your fingers. Don't handle it more than necessary, though; you want some of the gas bubbles to remain in the dough. It should look slightly blistered. Flour the peel and lift the disk onto the center. The dough is now ready to be topped. Shaping the Disk (Method 2): Take one ball of the dough and generously flour it, your hands, and the work surface. Then press it down and gently stretch it out to 6 to 8 inches. Supporting the disk with your knuckles toward the outer edge and lifting it above the work surface, keep stretching the dough by rotating it with your knuckles, gently pulling it wider and wider until the disk reaches 10 to 12 inches. Set the disk on a well-floured peel. It is now ready to be topped. If you don't intend to use the dough right away, wrap the balls individually in plastic and refrigerate for up to 3 days. Return to room temperature by leaving them out on the counter, covered in a damp cloth, for 2 to 3 hours before needed. Don't freeze the dough, but you can store it in the refrigerator, wrapped in plastic, for up to three days. In effect, when you're set to use it, you have your own ready-made dough.
Reprinted with permission from My Pizza by Jim Lahey, Clarkson Potter copyright (c) 2012