Cut the peeled pumpkin or squash sections into equal pieces about 5 or 6 cm in size and par-boil in lightly-salted water for about 8 minutes. Drain the pumpkin then cool under cold running water until cold.
Coarsely grate the pumpkin into a mixing bowl then add the oregano and season to taste. Mix gently (do not over mix or the pumpkin will not bind) then divide the mixture into 4 equal portions and form into round cake or patty shapes.
Heat the oil in a cast iron skillet then add the pumpkin cakes, pressing them down with the back of a spoon or a wooden spatula. After about 2 minutes carefully turn them over and cook for a further 2 minutes, or until they are nicely golden brown.