For the dough: Dissolve yeast in ½ cup warm water in a small bowl and set aside until foamy, about 10 minutes. Combine flour and salt in a large bowl. Add milk, ½ cup of the oil, 2 tbsp. warm water, and yeast mixture and stir until a dough forms. Turn dough out onto a lightly floured surface and knead until smooth, 10-15 minutes. Shape dough into a ball, dust with flour, then transfer to a large clean bowl. Cover bowl with a clean damp kitchen towel and set aside in a warm spot to rest until dough has doubled in bulk, 1-2 hours.
For the topping: Mix za'tar, sumac, and oil together in a medium bowl and set aside.
Preheat oven to 400°. Lightly grease a sheet pan with some vegetable oil and set aside. Turn dough out onto a clean surface, divide into thirds, and shape each piece of dough into a ball. Roll 1 dough ball out on a lightly floured surface into an 8" round and transfer to prepared sheet pan. Using your fingertips, make indentations all over the surface fo the dough, then brush with generous amount of topping. Bake flat bread until lightly browned and crisp around the edge, about 10 minutes. Repeat rolling out, indenting, topping, baking process with remaining dough balls and topping mixture, greasing baking sheet with more oil as needed. Serve warm or at room temperature.