In the bowl of a stand mixer, combine warm water, yeast, and sugar. Stir well and let stand for 10 minutes until it becomes foamy.
In a separate bowl, combine all purpose flour, bread flour, salt, and shortening. Use your hands to break up the shortening until no noticeable clumps remain.
Once the yeast is foamy, fit the mixer with the dough hook attachment and slowly add the flour on low speed. Pause to scrape down the sides as necessary.
After the dough comes together, place it on a lightly floured surface and knead until smooth and elastic (about 10 minutes). A good indicator for when it's done is by poking it with your finger: if the dough springs back quickly to fill the gap, you're good to go.
Coat a large bowl with the olive oil (3 tbsp is on the generous side). Place the dough in the bowl, then turn once to coat it in oil.
Cover loosely with a towel or plastic wrap and place in a warm spot, like next to a window with the sun beaming in.
Allow the dough to rise until doubled in size (about 1.5 hours).
Punch down the dough and place it on a lightly oiled baking sheet. Shape it into a flat, round disk about 9 inches across.
Sprinkle to top with sesame seeds and press them gently into the dough.
Loosely cover the dough and, once again, give it an hour to rise.
When the dough has risen a second time, whisk together the egg and 2 tbsp of cold water, then brush the top with the egg wash.
Place the dough in an oven at 425ºF for 10 minutes.
Turn the heat down to 375ºF and bake for an additional 25 minutes.
Recipe adapted from Nola Cuisine. Thanks, Danno!