Add the warm water, honey and yeast to the bowl of a stand mixer. (If you prefer, the dough can be mixed and kneaded by hand.) Whisk gently. Set aside for 5 minutes until foamy.
Add olive oil, salt and flour to the bowl. Using the paddle attachment, mix until combined. Switch to the dough hook and knead on low speed for 8 minutes, adding more flour 1 tablespoon at a time if the dough is very sticky. The dough should feel only slightly sticky when you touch it with your finger.
Cover the bowl with a clean kitchen towel and let rise for 45 minutes to 1 hour, until doubled in size.
Transfer the dough to a lightly floured work surface and knead it a few times to release any air bubbles. Shape the dough into a ball and cut it into 8 equal pieces. Working with one dough piece at a time, gently shape it into a ball and then use a rolling pin to roll it out into a 6-inch round. (If the dough springs back instead of rolling out well, let dough rest for 5 minutes and then try again - it will relax while it rests.)
Lay rolled out pita on a sheet of parchment paper and cover with a clean kitchen towel. (If you plan to bake the pita on baking sheets, place 4 pita on each of 2 parchment paper-lined baking sheets then cover.) Let dough rounds rest for 15 minutes. During this time the pita will puff up slightly.
To bake pita on a baking stone: Place baking stone on middle rack in oven and preheat oven to 450° F. Bake pita in batches (I can usually fit 2-3 on my baking stone), cooking for 2 minutes on the first side (until puffy) and then flipping and cooking for 1 minute on the second side, until pita are lightly golden brown in spots.
To bake pita on a baking sheet: Preheat oven to 425° F with a rack in the middle position. Line 2 baking sheets with parchment paper and place 4 pita on each baking sheet. Bake one pan of pita at a time, for 5-6 minutes on the first side, until puffy. Then flip pita over and bake for 2-3 minutes on the second side, until lightly golden brown in spots (check the bottoms, as they tend to brown faster than the tops). Repeat with remaining pita.
Wrap pitas loosely in a clean kitchen towel to keep them warm and soft until ready to serve.
Once cool, store leftover pita bread in a zip-top plastic bag at room temperature for up to 2 days or in the freezer for up to 3 months.