Heat olive oil in a skillet over medium-high heat.
Add leeks, salt, and pepper. Cook stirring for 7-8 minutes or until the leeks are softened.
Transfer the leeks in a bowl. Add pumpkin puree and stir to combine.
Make the Dough:
In a mixing bowl, whisk together flour, yeast, sugar and salt.
Add in water and olive oil.
Stir until the dough comes together. Start kneading dough until it is smooth. Cover the dough and let it rest for one hour.
Assemble the Empanadas
Roll out the dough to ⅛-inch thick on a slightly floured kitchen counter.
Cut out circles with empanada press. Gather the dough and roll it out again until it is all used. You should get 24 empanadas.
Place the circles of the dough on the empanada press. Brush the edges with some water. Fill each with a 1 tbsp of the pumpkin filling and place some crumbled feta on top.
Close the empanadas with an empanada press.
Preheat oven to 350F.
Arrange empanadas on two baking sheets. Brush each with beaten egg or olive oil.