Heat the oil in a 12-inch skillet over medium-high heat until shimmering. Add the Brussels sprouts, shallots, some of the salt, and pepper and cook until the sprouts begin to soften and the edges start to brown, 3 to 5 minutes.

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

Remove the skillet from the heat, stir in the peas, and cover; let the mixture sit until the peas are heated through, about 2 minutes.

Meanwhile, bring 4 quarts of water to a boil in a large pot. Add the pasta and the remaining tablespoon of salt and cook, stirring often, until al dente. Reserve some of the cooking water, then drain the pasta and return it to the pot.

Stir the Brussels sprout mixture, Parmesan, and walnuts into the pasta and toss to combine. Season with salt and pepper to taste, adding the reserved cooking water a splash at a time as needed to reach a glossy, creamy consistency, and serve immediately with extra Parmesan.