Lightly flour your work surface and rolling pin. Take one dough ball and roll it out into a thin, even sheet about 1mm thick, rotating the dough frequently to prevent sticking. Traditionally, an "oklava" (thin rolling pin) is used, but a standard rolling pin will work as well.
Place a generous amount of spinach filling over one half of the rolled-out dough. Fold the dough over the filling to form a half-moon shape, pressing the edges firmly to seal.
Preheat a griddle or non-stick skillet over medium heat. Place the filled dough onto the hot surface and cook for about 1 minute, or until the underside sets and small spots of color appear. Flip the gözleme and brush the cooked side lightly with butter or drizzle with oil. After another minute, flip again, brushing the other side. Continue flipping and cooking until both sides have golden brown spots and the dough is cooked through, around 4 to 5 minutes in total.
Repeat with the remaining dough balls and filling. To keep the cooked gözleme soft and moist, stack them in a pan and cover with a lid or a large plate until ready to serve.