Pat the Padrón peppers thoroughly dry and heat a tablespoon of olive oil in a frying pan until almost smoking. Toss the peppers in and let them sit for two to three minutes without stirring, until they get black blisters on one side. Then shake the pan a few times and cook for another minute. Sprinkle immediately with coarse sea salt and transfer to a plate.
Lower the heat and pour the remaining olive oil into the same pan. Crack the eggs into the pan and fry gently until the whites are set and the edges turn crispy brown, while the yolks remain completely runny. Spoon some of the hot oil over the yolks a few times while frying; this will set them slightly without hardening them.
Slide the eggs onto the plate next to the peppers. Shave the manchego into thin flakes over the dish using a vegetable peeler, and sprinkle everything with the smoked paprika powder and black pepper. The peppers provide almost no carbohydrates, while the eggs, sheep's cheese, and olive oil combine to create a breakfast consisting almost entirely of fat and protein.