Heat the oil in a large straight-sided skillet or braiser over medium-high heat. Add the sausage and cook, breaking it up with a wooden spoon to create small bite-size pieces, until browned and mostly cooked through, about 5 minutes.
Meanwhile, remove the rind from the Parmesan wedge and set aside. Grate about 1 cup of the cheese and set aside.
Add the tomatoes and Parmesan rind to the skillet with the sausage. Fill the empty tomato can with about 1/2 cup of water, swirl to remove any residual tomato in the can and stir the liquid into the skillet. Bring the mixture to a boil over medium-high heat, then reduce the heat to low, cover and simmer for 20 to 25 minutes.
Meanwhile, bring a large pot of salted water to a boil. Add the pasta and cook until al dente, about 12 minutes. Turn off the heat, stir in the frozen peas and let sit for about 1 minute. Reserve 1 cup of pasta water and then drain the pasta and peas.
Turn off the heat for the sauce, remove the Parmesan rind and discard. Transfer the pasta and peas to the sauce. Top the pasta with the reserved 1 cup of grated Parmesan and toss until the pasta is coated in the sauce. If the sauce feels dry, stir in a little pasta water in 1/4 cup increments until the sauce is silky and sticks to the pasta.
Serve the pasta topped with extra grated Parmesan.