In a large mixing bowl, combine the two flours and the salt.
In a measuring cup, measure out the lukewarm water and stir in the yeast. Allow this mixture to sit for a few minutes so that the yeast activates.
Create a well in the centre of the dry mixture, then pour in the wet mixture along with the olive oil. Use a wooden spoon to stir until the dough just starts to come together. At this point you will need to add more water. Add up to 1 cup extra and mix with your hands. You are looking for a smooth, slightly elastic dough, similar to pizza dough. Be sure not to make the dough too wet.
Gently knead the dough in the bowl for a few minutes. Remove the dough from the bowl and form it into a ball. Lightly flour the bowl, then place the dough back in it. Cover the bowl with a clean, damp dishcloth and allow the dough to rise at room temperature for 2 hours, or until doubled in size. While the dough is rising, make the garlic infused olive oil and eggplant puree.
Once the dough is done rising, separate it into four balls. Transfer the dough balls to a floured surface and use a rolling pin to roll the dough into the shape of a naan style bread. Repeat with the remaining dough.
Brush both sides of the flatbreads with the garlic infused oil and sprinkle with za'atar.
Place the flatbreads on the heated grill and cook, with the lid closed, for about five minutes per side.