Dissolve the yeast in 3/4 cup warm water. Stir in the sugar and set aside the mixture until it blooms with a slightly foamy top, 5 to 10 minutes.
Sift together the all-purpose flour and white whole-wheat flour in the bowl of a stand mixer. Make a deep depression in the middle with a wooden spoon and add the olive oil. Slowly pour in the bloomed yeast mixture.
Using the dough hook attachment, knead the dough on low speed until the mixture starts to come together. Slowly pour in 1 1/2 cups of warm water, adding up to an additional 1/4 cup if needed so the dough looks smooth and feels slightly sticky. Add the salt, increase the speed to medium and continue to knead until a soft and stretchy dough forms (it should not feel sticky), 10 to 12 minutes. Transfer the dough to a large oiled bowl and cover with a damp towel. Allow it to rise in a dark area until it doubles in size, about 1 hour.
Transfer the risen dough to a lightly floured work surface and gently punch it down. Divide the dough into 18 even pieces, and form these pieces into smooth balls. Using a rolling pin, roll a piece of dough into circle that's about 5 inches wide and 1/2 inch thick. Repeat with the remaining dough pieces. Cover the dough with the damp towel and let it rest for 20 minutes.
Meanwhile, place a nonstick baking sheet in the oven and preheat the oven to 525 degrees F.
Remove the baking sheet from the oven and carefully transfer 3 pieces of dough to it; lightly spritz them with water from a spray bottle. Bake the pitas until they’re puffed up and light brown with a few dark brown freckles in places, watching very closely because they burn quickly, 4 to 5 minutes.
Remove the pitas from the baking sheet and allow them to cool on plates or a wire rack covered in paper napkins to maintain softness. Place the baking sheet back in the oven for 5 minutes. Repeat with the remaining pitas.
Serve the pitas immediately with Tahini Sauce, if desired. Store them in resealable bags at room temperature for up to 2 days or freeze them in resealable bags for up to 3 months.
Combine the lemon juice, garlic, salt and cumin in a large bowl. Add the tahini to the garlic mixture, stirring with a whisk or fork to combine (the mixture will be very thick). Pour in 1/2 cup cold water, stirring until incorporated; feel free to add more until you reach the desired consistency. Garnish with chopped parsley, if desired.