Bring a large pot of lightly salted water to a boil. Add the pasta and cook until just tender, about 8 to 10 minutes. While the pasta cooks, place the cream, ham, garlic and oregano in wide skillet set over medium heat. Bring the cream to a simmer, season with salt and pepper, and then reduce the heat to low.
When the pasta is cooked, reserve ¾ cup of its cooking liquid, then drain the pasta well. Add the pasta and reserved cooking liquid, peas and 1/3 cup of cheese to the skillet and toss to combine. Spoon the pasta into bowls and serve with additional Parmesan cheese, if desired.