Heat the oil in a 12-inch skillet over medium-high heat until it shimmers but does not smoke. Add the Brussels sprouts, shallots, salt, and pepper and cook until the sprouts begin to soften, 3 to 5 minutes.

Stir in the broth and cream, cover the skillet, and simmer until the sprouts are tender, about 3 minutes.

Remove the skillet from the heat, stir in the peas, and cover again. Let the mixture sit for about 2 minutes to heat the peas through.

While the vegetables are simmering, bring 4 quarts of water to a boil in a large pot. Add the pasta and the remaining salt and cook, stirring often, until the pasta is al dente.

Reserve 1/2 cup of the pasta cooking water, then drain the pasta and return it to the pot.

Add the Brussels sprout mixture, Parmesan, and walnuts to the pot and toss to combine. Season with additional salt and pepper to taste, adding the reserved cooking water as needed to reach your preferred consistency.

Serve immediately with extra Parmesan passed at the table.