Place the pita halves, cut-side up, on a baking sheet. Brush with the oil and sprinkle with 3/4 teaspoon of the paprika; set aside.
In a large skillet, over medium heat, melt the butter. Add the spinach, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper and cook, tossing frequently, until the spinach has wilted, about 2 minutes.
Divide the spinach among the pitas. Make a well in the center of each bed of spinach and crack 1 egg into the well. Season the egg with the remaining paprika, salt, and pepper and sprinkle with the Feta. Bake until the eggs are set, about 11 minutes for runny yolks. Serve with the mint and stuffed grape leaves (if desired).