Make the dough: Whisk the vanilla into the milk in the bowl of a stand mixer fitted with the dough hook. Use a fork or your fingers to lightly mix the yeast into the milk; if using active dry yeast, stir the yeast into the milk. Then, in this order, add the flours, eggs, sugar, salt, and finally the butter in small pinches.

Mix on the lowest speed, stopping the mixer to scrape down the sides and the bottom of the bowl as needed and to pull the dough off the hook as it accumulates. There and break it apart so it mixes evenly, until the dough is well combined, about 2 minutes (it will not be smooth). If the dough is very dry, add more milk, 1 tablespoon at a time; if the dough looks wet, add more all-purpose flour, 1 tablespoon at a time, until the dough comes together. Increase the mixer speed to medium and mix until the dough is smooth and has good elasticity, 4 minutes.

Stretch and fold the dough: Lightly dust your work surface with flour and turn the dough out on top; lightly dust the top of the dough and the interior of a large bowl with flour. Grab the top portion of the dough and stretch it away from you, tearing the dough. Then fold it on top of the middle of the dough. Give the dough a quarter turn and repeat the stretch, tear, and fold. Continue to do this until you can stretch a small piece of dough very thin without tearing it, about 5 minutes. Then use your hands to push and pull the dough against the work surface and around in a circular motion to create a nice round of dough. Set the ball in the floured bowl, cover the bowl with plastic wrap, and set it aside at room temperature for 30 minutes.

Chill the dough: Set the dough on a piece of plastic wrap and press it into a 1-inch-thick rectangle. Wrap the dough in plastic wrap and refrigerate it for at least 1 hour or up to 24 hours before proceeding with one of the recipes in this chapter.