Step 1Heat the oven to 400 degrees. Brush a 13-inch oval gratin or baking dish with butter.
Step 2In a large, shallow saucepan, bring the chicken broth to a boil. Add the chard, cover and simmer until the chard wilts, about 4 minutes. Remove from heat. Using tongs, remove the chard and drain well, reserving the broth. When the chard is cool, gently squeeze excess moisture out of the chard back into the saucepan. You should have about 1/2 cup broth. If you have too much, boil briefly to reduce to 1/2 cup or add additional broth to equal 1/2 cup broth. Add the cream, milk and salt to the broth and set aside.
Step 3Arrange 1/3 of the parsnip slices in the prepared baking dish. Grate nutmeg generously over the parsnips, season with pepper and sprinkle with 2 tablespoons cheese. Scatter half of the chard over the cheese. Arrange half of the remaining parsnips atop the chard. Grate the nutmeg generously over the parsnips, season with pepper and top with the remaining chard. Sprinkle 2 tablespoons cheese over the chard and top with the remaining parsnips. Bring the broth mixture to a simmer and pour it evenly over the parsnips. Sprinkle with the remaining 1/4 cup cheese. Grate nutmeg generously over the cheese and season with pepper. Cover the gratin with foil and bake until the parsnips are just tender when pierced with a skewer, about 50 minutes. Uncover and continue to bake until the top of the gratin is evenly browned and the liquid is reduced and bubbles very thickly at the edges, about 30 minutes. Cool slightly and serve.