Cook and stir the onion in oil in a 10-inch skillet until tender but not brown. Add the sausage, cooking and stirring, until the sausage is done then drain. Mix the eggs, half-and-half and the oregano well with a fork. Pour into the sausage mixture in the skillet. Cook over medium heat. As the mixture begins to set at the bottom and sides, gently lift the cooked portion with a fork or spatula so that the uncooked egg mixture can run under the cooked portion. Avoid constant stirring, as this will make the eggs tough and hard. Cook until all the eggs are cooked but still moist, about 3 to 5 minutes.