Sift flour into a large bowl. Slowly add water, a spoon at a time and 2 tablespoons oil. Mix to form a soft dough. Turn dough onto a lightly floured surface. Knead for 5 minutes or until elastic. Cut dough into 4. Place on a greased baking tray. Cover with a clean tea towel. Stand in a warm, sunny place for 20 minutes to rest the dough.
Roll each piece dough into a 35cm x 45cm rectangle. Place one-quarter of spinach over half of each rectangle. Top with feta and season with salt and pepper. Fold dough over to enclose filling. Press edges together to seal.
Preheat a barbecue plate on medium-high heat. Brush one side of each gozleme with 2 teaspoons oil. Cook for 2 to 3 minutes or until base is golden. Brush uncooked side with remaining oil. Turn over and cook for 2 to 3 minutes or until golden and crisp. Remove to a serving plate. Cut gozleme into quarters. Serve with lemon wedges.