Create a sponge by combining the eggs, water, sugar, 3 cups of the flour, and yeast. Mix until well blended, then set aside.

With an electric mixer, stand mixer (with paddle attachment), or food processor, mix the 1 ⅞ cups butter with the ¾ teaspoon salt and the ½ cup flour. Use very low speed to avoid incorporating air into your butter. Mix until there are no solid lumps of butter left. Scrape butter onto a piece of lightly floured plastic wrap and use another piece of plastic wrap to help smooth the butter into an 8x8 inch square. Use the sides of the plastic wrap to help even up the edges while you do this. Wrap butter and place it on a flat surface in the refrigerator for at least 30 minutes to allow it to solidify.

In your stand mixer, combine the sponge with the tablespoon salt, vanilla extract, melted butter, and another 2 ½ cups of flour. Use your dough hook to knead the dough for 4-5 minutes, until it is smooth and elastic. It will be soft and tacky, but definitely not sticky. Add flour as necessary. The dough will barely touch the bottom of the bowl once you're done with it. Turn dough out onto a lightly floured surface, form it into a square, and wrap and store it in the fridge for 30 minutes.

Remove the chilled dough from the fridge and roll it into a 12x12 inch square. Unwrap the square of butter and place it in the center of the dough like a diamond. Fold in the corners of dough over the butter. Dab a little water on your fingers and pinch together the seams of the dough to seal.

Check to make sure the dough isn't sticking on bottom; dust with flour if necessary. Dust top of dough lightly with flour, then roll out to a 10x20 inch rectangle. Gently brush off any excess flour if necessary. Make a tri-fold with the dough; fold one third of the dough up toward the center. Line up the edges, and use wet fingers to tack the seams in place. Repeat with the other third of the dough. Turn the dough 90 degrees, then repeat the entire rolling and folding process.

After you've rolled and folded the dough for the second time, wrap dough and place in fridge for 30 minutes to allow the dough to rest and the butter to chill.

Remove the dough from the refrigerator. Place upside down and roll out to a 10x20 inch rectangle again. Fold into a trifold again, lining up seams and tacking them into place, and turn dough 90 degrees. Repeat the rolling and folding process for a fourth time, then wrap the dough well and refrigerate it overnight.

The next day, remove the dough from the fridge and cut it into half. Return one half to the refrigerator while you work. Place the other half on a lightly floured surface and roll into a 12x18 inch rectangle. Use a ruler and a pizza cutter or pastry blade to trim the edges to make the dough a perfect rectangle and to cut off any sealed edges.

Add fillings (like chocolate) in a row across the dough just above the notch, if desired. You can also sprinkle cinnamon sugar on the dough if you'd like.

Starting at the short edge, pull gently on the corners to separate the notch, then roll up the dough. The tip of the dough should end up on the bottom of the croissant; stretch the dough if necessary to get this to happen. Use your fingers to wet the tip to tack it into place. Curve in the ends of the croissant toward the center. Place croissants on a parchment-lined baking sheet and refrigerate for 30 minutes, or freeze if you won't use them immediately. (If freezing, the original recipe says to thaw in the fridge overnight prior to baking, but I've also seen that you can bake straight from the freezer - extending the baking time, of course.)

Repeat rolling and shaping process with the other half of the dough.

Heat the oven to 425F. Beat one egg with one tablespoon water, and brush on croissants. Bake croissants for 15 minutes, then reduce the temperature to 350F and bake for another 15 minutes. Croissants should be a deep golden brown, even in the seams, when done. Cool briefly, but serve warm if possible.