In the bowl or stand mixer, combine the lukewarm water, yeast, sugar, and 2 cups of all-purpose flour, mixing everything together really well. Cover with a towel and keep in a warm spot for 10 minutes.
Next, add olive oil, remaining 3 cups of flour, milk powder, salt, za'atar and vinegar. Stir together very well. Using a hook attachment, knead the mixture 6-8 minutes until it comes together nicely as a dough. It should be smooth and not sticky. If it is sticky, add a tbsp of flour at a time.
Shape the dough into a ball and place it back in the bowl, now coated with olive oil. Cover and allow to proof in a warm spot for 30 minutes. it should double in size.
Preheat the oven to 375 F.
Then, form the dough into golf ball sized balls and stuff with a cube or tsp of cheese mixture.
Place in a 12-inch cake pan with about an inch of space between them as they will exaond when baking.
Beat the egg and milk together for the egg wash. Brush the egg wash over the top of the buns and sprinkle za'atar on top of each.
Bake for 20-25 minutes until they are golden brown. Enjoy warm or room temperature.