Heat oven to 425° F. Place the pita halves, cut-side up, on a baking sheet. Brush with the oil and sprinkle with ¾ teaspoon of the paprika; set aside.
In a skillet, over medium heat, melt butter.
Add the spinach, ½ teaspoon salt, and ¼ teaspoon pepper and cook, tossing frequently, until the spinach has wilted, about 2 minutes.
Divide the spinach among pitas. Make a well in the center of each bed of spinach and crack 1 egg into it. Sprinkle the egg with the remaining paprika, salt, and pepper and the Feta.
Bake until the eggs are set, about 11 minutes for runny yolks. Serve with the mint and stuffed grape leaves (if desired).