If dough is refrigerated, let stand at room temperature, covered, for 30 minutes.
Preheat oven to 450 degrees F (230 degrees C) and put a pizza stone on the bottom rack.
Halve dough and roll each half into a 10-inch oval on a lightly floured surface. Brush each oval with 1 tablespoon oil. Put 1 oval on a cornmeal-dusted baking sheet or baking peel and transfer to the pizza stone.
Bake until golden brown, about 10 minutes. Repeat with remaining oval. Cut flatbreads into wedges or tear into pieces.
Stir together yogurt, chevre, parsley, garlic, lemon zest, and dill in a bowl.
Put 3 tablespoons oil in a small bowl and dukkah in another bowl. Serve flatbread with all 3 bowls. Dip flatbread in oil, then in dukkah, and use chevre-yogurt dip for spreading.