Heat a saucepan over medium heat. Add the oil and 1 tablespoon of the butter and cook until foaming. Add the onions, season with a little salt and saute about 5 minutes. Add the garlic and cook 1 minute more. Next, add the mushrooms and some salt and pepper and cook until the mushrooms have released all their liquid, about 5 minutes more. Add the peas and cook until heated through, another 2 minutes. Keep warm.
Meanwhile, bring a pot of water to a boil and salt generously. Cook the cavatelli pasta to al dente, 4 to 5 minutes for fresh, or according to the package instructions. Set a strainer over a large heatproof measuring cup in the sink and carefully drain the pasta, reserving 2 cups of pasta water.
Heat a large saute pan over medium heat and add the remaining 3 tablespoons butter. When melted, add the Pecorino Romano and 1 cup pasta water, whisking vigorously. Whisk in the cream. Add the pasta and toss to coat. Season with salt and add additional pasta water if the sauce is too thick. Stir in the mushroom mixture.
Serve immediately, garnishing with additional freshly ground black pepper and freshly grated cheese.
Combine the 00 and semolina flours with the salt in a food processor fitted with a metal blade; pulse a couple times. Add the olive oil and 1 cup room-temperature water to the flour mixture and pulse a few more times. Mix until a shaggy mass forms, then turn out onto a clean surface. Knead vigorously until the dough is smooth, elastic and bounces back when pressed, about 10 minutes.
Cut the dough in half and work with one piece at a time (cover the other piece with a clean, damp kitchen towel). Using a bench scraper or knife, cut a walnut-sized piece of dough and roll it into a thin rope about 1/3 inch in diameter. Using the 4 fingers of your hand (no thumb), roll the rope of dough back and forth until it is a long, slender, tube-like shape (but not enclosed like a tube). Halve the pieces of pasta. Set aside on a clean towel sprinkled with semolina flour. Repeat with the remaining dough. Rest the pasta for 30 minutes.
To cook, generously salt a large pot of boiling water. Add the cavatelli and cook until al dente, 4 to 5 minutes. Drain and use with your sauce recipe of choice.