Preheat your oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish with olive oil. Thoroughly wash the sliced leeks in a bowl of cold water to remove any trapped dirt, then drain them well in a colander.

Heat the extra virgin olive oil in a large skillet over medium heat. Add the sliced leeks and sauté for 8 to 10 minutes, stirring occasionally, until they are soft, sweet, and translucent.

Stir the rinsed white rice, sweet Hungarian paprika, dried mint, salt, and black pepper into the skillet with the leeks. Cook for another 2 minutes, stirring constantly to toast the rice slightly and bloom the spices.

Transfer the leek and rice mixture into the prepared baking dish, spreading it out into an even layer. Pour the warm vegetable broth over the mixture and gently stir to ensure the rice is completely submerged.

Cover the baking dish tightly with aluminum foil and bake in the preheated oven for 30 minutes. Remove the foil and bake uncovered for an additional 15 minutes, or until the liquid is fully absorbed and the top of the rice is slightly golden.

Let the casserole rest for 5 minutes at room temperature before serving to allow the flavors to meld. Garnish the top with crumbled feta cheese and serve warm alongside a fresh cucumber salad.