In a large bowl, whisk together the bread flour, 1 ¾ cups all-purpose flour, yeast, and sugar until evenly combined. Make a well in the center and add the water, Greek yogurt, olive oil, and salt. Mix the ingredients until a dough forms, then turn it out onto a clean work surface.
Knead the dough for 8-10 minutes until extremely soft and supple. Alternatively, you may use a stand mixer on low fitted with the dough hook attachment (keep in mind that the kneading time may shrink). If the dough becomes too sticky while kneading, add more all-purpose flour as needed (you may need to add up to ½ cup). The dough is finished kneading if it is tacky* and springs back when poked.
Lightly oil a bowl and place the dough inside. Cover the bowl and let the bread proof for 60-90 minutes, until doubled in size (this may happen slower or faster, depending on how warm your kitchen is). Test if it is ready by poking it; if the dough springs back and leaves a slight indentation, it's ready to be shaped. If it springs back with no indentation, it needs longer. If it doesn't spring back at all, it's overproofed.
Punch the dough down and turn it out onto a work surface (not floured), and divide it into 8-10 pieces, depending on how large you want your pitas.
Pinch the edges of the pitas together to form a teardrop shape. Place each ball seam-side down onto the counter. Loosely cup your hand over a piece of dough and move it in rapid circles against the surface of the table, until a neat ball is formed.**