Step 1
Combine warm milk yeast and sugar in the bowl of a stand mixer fitted with the paddle attachment. Let stand until mixture is foamy 5 to 10 minutes.
Step 2
Beat milk mixture on low speed. Add 1 cup flour beating just until combined. Add melted butter 2 tablespoons olive oil and 1/2 cup flour; beat until combined. Beat in eggs. Add remaining 1 3/4 cups flour and salt. Beat until a soft sticky dough forms.
Step 3
Spray a large bowl with cooking spray; add dough. Cover loosely and let rise until doubled in size about 1 hour.
Step 4
Preheat oven to 350 degrees F (175 degrees C). Grease a 12-cup muffin tin with cooking spray.
Step 5
Punch dough down gently. Pinch off 36 small pieces of dough and roll into balls.
Step 6
Whisk 6 tablespoons za'atar and 1/3 cup olive oil together in a bowl. Immerse dough balls in oil mixture to coat all sides. Arrange 3 dough balls in each muffin cup. Sprinkle remaining 2 tablespoons za'atar over dough.
Step 7
Cover muffin tin loosely with greased plastic wrap. Let rise until dough balls puff up over the rim of the tin about 25 minutes.
Step 8
Bake in the preheated oven until puffed up and golden about 25 minutes. Cool for 5 minutes in the pan. Lift out of the pan and cool briefly on a wire rack about 5 minutes more.