Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain.
Melt 2 tablespoons butter in a deep frying pan over low heat. Cook onions, chard stalks, and garlic until soft while stirring, 5 to 10 minutes. Add chard leaves and the rest of the butter. Cover and cook until chard is soft and wilted, about 8 to 10 minutes.
Combine cream and mustard in a bowl; season with salt and pepper. Add to chard in pan and simmer for 3 to 5 minutes. Add tomatoes, cover, and cook until tomatoes are soft, about 5 minutes. Season with nutmeg, salt, and pepper.
Add drained pasta to vegetable sauce and stir to combine. Serve with Parmesan cheese.