Bring a large pot of water to a boil for the pasta.
In a small skillet, toast the walnuts over medium heat, stirring often, until fragrant, about 5 minutes. Transfer to a plate; let cool.
In a food processor, pulse the chard, lemon juice, garlic and walnuts until the mixture is finely chopped. Season with nutmeg and salt and pepper. With the machine running, stream in the EVOO and process until a smooth pesto forms. Transfer the pesto to a large serving bowl; stir in the grated cheese.
Salt the boiling water, add the pasta and cook to al dente. Reserve 1 cup of starchy cooking water, then drain the pasta. Stir the cooking water into the pesto. Add the pasta to the serving bowl and toss with the pesto. Season; top with the shaved cheese.