Make the dough: In a large bowl, whisk together the all-purpose flour, yeast, and fine sea salt. Add the cool water and, using your hands, mix everything together until you don’t see any flour.
Divide the dough in half. Transfer the first half to a food processor fitted with the dough blade and process for about 1 minute until the dough is warm—but not hot—to the touch. Set the first half aside and process the second half for about 1 minute. Repeat this procedure four times, waiting 1 to 2 minutes between kneadings to allow the food processor to cool down.
Transfer the kneaded dough halves back to the large bowl that you first used to mix the ingredients together. Cover it with plastic wrap or a kitchen towel and let the dough rest at room temperature for 1 hour and 15 minutes.
Give the dough a quick knead just to combine the halves, then, using a bench scraper, divide it into 3 equal pieces. Gently round each piece into a ball and set all 3 on a lightly oiled parchment paper–lined baking sheet. Cover the dough loosely with plastic wrap and refrigerate overnight or for at least 8 hours. (The dough will keep in the refrigerator for up to 3 days.)
Preheat the oven to 500°F with a 10-inch cast-iron skillet set on a rack in the middle of the oven. Heat the pan for a full 30 minutes.
Meanwhile, remove one of the pieces of dough from the refrigerator and let it warm up for 15 to 20 minutes. Dust your work surface with coarse whole-wheat flour and place the ball of dough on the dusted surface.
With your index and middle fingers, begin to punch down the dough, starting in the center of the ball and working out. Create a rim about 1-inch-wide by dimpling the dough with your fingers. Stretch the dough by lifting it up from its center with your fist (just like you’d shape pizza dough). The stretched bialy should be 8 inches across. Spread 2⁄3 cup of the onion topping over the dimpled part of the bialy. Sprinkle 2 teaspoons of the bread crumbs and ¼ teaspoon of the flake sea salt on top.
Carefully pull the hot skillet out of the oven and place it on a heatproof surface. Using your hands or one hand and a large spatula, gently scoop underneath the bialy, trying not to disturb its topping. Quickly transfer it to the skillet and return the pan to the oven. Reduce the oven temperature to 450°F and bake for 15 to 20 minutes, until the crust is golden brown.
Remove the skillet from the oven and use a spatula to transfer the bialy to a wire rack. Repeat to make 2 additional bialys. The second and third bialys can be made right away, or, as noted above, the dough can be kept in the refrigerator for up to 3 days.