Mix water, yeast, salt, and sugar together in a large bowl. Add all-purpose flour, whole wheat flour, olive oil, and corn oil; mix using your hands, adding more whole wheat flour if needed, until dough holds together. Mix za’atar into dough until evenly incorporated (see notes).
Shape dough, about 1/4 cup per piece, into rounds on a floured work surface. Arrange rounds on baking sheets.
Bake in the preheated oven until lightly browned, about 25 minutes.