Add the lukewarm water to the bowl of a stand mixer fitted with the dough hook. Whisk in the yeast and honey and let sit 5 minutes, until foamy. (If it doesn’t foam, your yeast may be dead or the water too hot or cold.)
Add the flours and salt. Mix on low until the flour is incorporated, then increase to medium-high for 5 minutes. (If your mixer struggles or walks across the counter, drop to medium.)
Turn off the mixer, cover the bowl with plastic wrap, and rest for 10 minutes.
Resume mixing on medium-high for another 15 minutes. The dough will be very wet and sticky, closer to a thick batter than a traditional dough. That’s correct, and it’s what gives the crust its open, airy crumb. Don’t add more flour.
Pour ¼ cup olive oil into a large bowl. Scrape the dough into the bowl with a rubber spatula and coat the top in oil. Cover and let rise until doubled in size, [about 1–1½ hours depending on your kitchen’s temperature].
Add the remaining ¼ cup oil to the 9x13 or 10x15 pan. Pour the dough into the pan and gently pat it into the corners with oiled fingers. If it springs back, let it rest 10 minutes and try again. Cover and let rise another 30 minutes, until puffy.
Dimple the dough heavily with your fingers, pressing all the way to the bottom of the pan.