Stir all of the ingredients together in a medium bowl until combined.
Cover with plastic wrap and let sit at room temperature. Let it rise for about 2 hours then wait for it to fall (about 4 hours) before using.
For the Dough
Combine 3 cups of the flour and the yeast in a standing mixer fitted with the dough hook. With the mixer on low speed, add the water and mix until the dough comes together, about 2 minutes. Stop the mixer, cover the bowl with plastic wrap (no need to remove it from the mixer), and let sit at room temperature for 20 minutes.
Remove the plastic wrap, add the sponge and salt, and knead the dough on medium-low speed until it is smooth and elastic, about 8 minutes. If after 4 minutes more flour is needed, add the remaining ½ cup flour, 2 tablespoons at a time, until the dough clears the sides of the bowl, but sticks to the bottom.
Turn the dough out onto a lightly floured counter and knead by hand to form a smooth, round ball. Stretch a small piece of dough out thin. If the dough does not tear and you can see light through the dough, it is adequately kneaded. (If the dough tears, knead a bit more and test again.) Place the dough in a large, lightly oiled bowl and cover tightly with greased plastic wrap. Let rise in a warm place until doubled in size, about 1 hour.
Slide a plastic bench scraper or rubber spatula under one side of the dough; gently lift and fold a third of the dough toward the center. Repeat the previous step with the opposite side of the dough. Finally, fold the dough in half, perpendicular to the first folds. The dough shape should be a rough square. Cover, let rise for 30 more minutes, then repeat the turning process. Cover and let rise until the dough has doubled in size, about 30 minutes longer.
Top a large rimless (or upside down) baking sheet with parchment paper. Turn the dough out onto a lightly floured counter and divide it into 2 equal pieces. Working with one piece of dough at a time, gently pat the dough into a rough 8 by 6-inch rectangle, with the long edge towards you. Gently fold the bottom third of the dough up to the center and press to seal. Gently fold the bottom of the dough up to the top and press to seal. Using the side of your floured hand, create a large crease down the middle of the dough. Working from one end of the loaf to the other, pinch the top and bottom of the dough together securely. Gently roll the baguette back and forth to stretch it into an even 15-inch-long baguette, about 2½ inches wide. Lay the loaf seam side down on the parchment-lined baking sheet, spaced about 5 inches apart. Mist the baguettes with vegetable oil spray, cover loosely with plastic wrap, and let rise in a warm place until nearly doubled in size and the dough barely springs back when poked with knuckle, 1 to 1½ hours.
Meanwhile, adjust an oven rack to the lower-middle position, place a baking stone on the rack, and heat the oven to 500 degrees. Place a baking sheet, which will serve as the steam pan, with a 1-inch rim under the shelf with the baking stone. Let the baking stone heat for at least 30 minutes (but no longer than 1 hour).
For the glaze and to bake
Remove plastic wrap from the dough 15 minutes prior to baking. Just before baking, score the top of the breads with a razor blade or sharp knife, cutting four ½-inch-deep slashes along the width of each baguette. Brush the breads with the egg-water mixture, then spray lightly with water. Carefully slide the breads and parchment onto the hot baking stone. Immediately reduce the oven temperature to 425 degrees and pour 1 cup hot water into the steam pan. (Always use an oven mitt and wear long sleeves when adding water to the hot steam pan to prevent steam burns. Using a watering can with a long spout when pouring the water into the steam pan provides control and distance from the hot steam.) Bake until the crust is deep golden brown and the center of the bread registers 210 degrees on an instant-read thermometer, about 25 minutes, rotating the loaves halfway through baking.
Transfer the breads to a wire rack, discard the parchment, and let cool for about 30 minutes before serving. (Letting it cool allows the crust to become more crispy).