Combine the sugar, yeast, lukewarm water, and milk in a bowl. Set aside for 5 minutes, until foamy. Stir in the olive oil.
In a large bowl, combine the flour, salt and thyme.
Gradually add the yeast mixture to the flour, mixing after each addition, until a dough forms.
Turn the dough out onto a lightly floured surface and knead for 4–5 minutes, until smooth.
Brush a bowl with olive oil, add the dough and turn to coat. Cover with a damp towel and let rise in a warm spot for 40 minutes to 1 hour, until doubled.
Cut the dough into 8 equal pieces and shape each into a ball. (new step)
Flatten each ball and gently stretch it with your hands into a 7–8 inch round.
Heat a cast-iron skillet over high heat and add 1 teaspoon olive oil.
Cook one pita at a time for about 1 minute per side, until it puffs slightly and browns in spots. Add a little oil after every 1–2 pitas. Stack the cooked pitas in a clean kitchen towel to keep them soft.