Soak the dried mushrooms in 1 cup boiling water until they soften, about 10 minutes. Heat the chicken broth in a medium saucepan over low heat. Transfer the mushrooms to a cutting board and chop very coarsely. Slowly pour off the mushroom soaking liquid into the chicken broth, leaving any sediment behind.
Heat the oil in a large saucepan over medium-high heat for 30 seconds. Add the onion; season generously with salt and pepper. Cook, stirring, until the onion is soft and translucent, about 3 minutes. Add the pasta and cook, stirring, until it browns in places and is coated evenly with the oil, about 1 minute. Add the mushrooms and 1 cup of the broth; stir well. When the broth reaches a boil, reduce the heat so that it simmers gently. When the broth is almost completely absorbed, continue to add broth in 1/2-cup increments, stirring until it’s almost completely absorbed before adding more. After the third addition is absorbed, toss in the peas and continue adding broth and stirring until the pasta is just tender, about 14 minutes total (you may not need to use all of the broth). Remove the pan from the heat. Stir in the Parmigiano, butter, half of the lemon zest, and half of the basil. Taste and add more lemon zest, salt, or pepper if you like. Serve immediately with the remaining basil sprinkled on top.