Combine oil and red pepper flakes in a small skillet and cook over low heat for 3-4 minutes. Stir in paprika and turn off the heat. Set aside to cool slightly.
Meanwhile, combine yogurt with garlic and season with salt and pepper to taste. Spoon it into a serving dish and spread flat to create a bed for eggs.
Fill a medium size pan with water and bring to a boil. Stir water with a spoon to create a whirlpool and crack in an egg. Cook until the white is just set and yolk is still runny, about 2-3 minutes.
Lift the egg out of water with a slotted spoon and place it on yogurt bed. Repeat, if using another egg.
Spoon red pepper oil over egg and serve immediately with crusty gluten-free bread.
Notes
Note: If you could add a sixth ingredient to this recipe, make it tomatoes and sauté them along with onions and peppers to make a flavorful ragout that is used as the base for various dishes throughout the Mediterranean region. If you are avoiding eggs, definitely use tomatoes instead!