Warm the chicken broth in a small saucepan. Remove from the heat and add the dried cranberries; set aside to plump.
Meanwhile, heat 3 tablespoons olive oil in a large pot over medium heat. Add the red onion and chard stems and cook, stirring, until the stems are just tender, 10 to 12 minutes. Add the nutmeg, 1 teaspoon salt and a few grinds of pepper. Add the chicken broth and dried cranberries. Add the chard leaves, pressing them down to fit, if needed. Cover and simmer until the leaves just start to wilt, 1 to 2 minutes. Uncover and increase the heat to medium high. Cook until the leaves are just tender, 3 to 5 minutes. Season with salt.
Transfer the chard mixture to a serving bowl with tongs, leaving the liquid in the pot. Stir the remaining 1 tablespoon olive oil into the liquid and increase the heat to high. Bring to a boil and cook until the liquid is syrupy, about 2 minutes. Pour over the chard.