Heat butter in a skillet over medium-low heat. Gently crack eggs into pan and cook, undisturbed, just until whites turn opaque, about 2 minutes. Season eggs with salt and cayenne pepper; flip, taking care to not break yolks, and cook until set, about 30 seconds.
Cook chorizo in a skillet over medium-high heat until browned, about 10 minutes.
Arrange potato chips on a plate and lay eggs over top. Cover eggs and chips with chorizo and top with Cheddar cheese and oregano.