Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
In a large bowl, combine the flour, baking powder, salt, and black pepper. Stir until well mixed.
Create a well in the center of the dry ingredients and add the pumpkin puree and olive oil. Mix until a dough forms.
Lightly flour a clean surface and turn the dough out onto it. Knead gently for about 1.5 minutes, just until the dough is smooth and not sticky.
Divide the dough into two equal portions. Roll each portion out into an oval or round shape about 0.5 inches thick. Transfer them onto the prepared baking sheet.
Brush the top of each flatbread lightly with olive oil. Sprinkle dried oregano evenly over each, then scatter crumbled feta cheese on top.
Place the flatbreads in the preheated oven and bake for 17.5 minutes or until the edges are golden brown and the feta is slightly melted.
Remove from the oven and let cool for a minute. Garnish with chopped fresh parsley and sprinkle red pepper flakes if desired. Cut into wedges and serve warm.