Boil the broth and mushrooms over medium-high heat in a small saucepan. Turn off the heat and allow the mushrooms to soften in the liquid, about 20 minutes. Remove the mushrooms and coarsely chop. Reserve the liquid.
Bring a large pot of salted water to a boil over high heat. Cook the pasta until tender but still firm to the bite, stirring occasionally, 8 to 10 minutes. Drain and reserve about 1 cup pasta water.
Heat the oil over medium-high heat in a large nonstick saute pan. Add the shallots mushrooms, 1 teaspoon salt and 1/2 teaspoon pepper. Cook until the shallots are soft, about 3 minutes. Add the chard in batches and cook until wilted, about 7 minutes. Sprinkle 1 teaspoon salt and the remaining 1/2 teaspoon pepper.
Remove the pan from the heat and add the mascarpone cheese. Stir until the mixture is smooth and creamy. Add the mushroom mixture, chicken and the reserved mushroom-cooking liquid to the pasta. Add reserved pasta water if necessary. Toss until all the ingredients are coated and serve.