In a large bowl, whisk the eggs, season with salt and pepper and add the oregano. In a nonstick pan, cook 1 tablespoon of the butter until it foams and subsides. Pour 1/4 of the egg mixture into the pan and cook until it has solidified and is easily slid from the pan, about 2 minutes. Repeat this procedure 3 times with the remaining butter and egg mixture, setting each omelet aside to cool.
When the omelets are cool, cut each into 1/4-inch thick strips.
In a medium, heavy-bottomed saucepan, combine the tomato sauce and the chilis and bring to a boil, then reduce heat to a simmer. Add the egg strips and simmer for 10 minutes. Top with the grated Pecorino and serve.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic, and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot, and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.