Preheat the broiler to high.
In a large bowl, mix together the breadcrumbs, milk and 1 tablespoon Marsala. Leave to soak for 5 minutes. Add the chicken, pecorino, egg, parsley, 1/2 teaspoon of the salt and the pepper. With your hands, gently mix together the ingredients until just combined. Form the mixture into tablespoon-size balls and place on an oiled baking sheet. Broil for 5 minutes, or until the meatballs are beginning to brown and are just barely cooked through. Remove from the oven and set aside.
In a straight-sided skillet, heat 1 tablespoon of the olive oil over medium-high heat. Add the mushrooms and cook, stirring with a wooden spoon, until the mushrooms are brown on all sides, about 5 minutes. Add the shallots and the remaining 1/2 teaspoon salt and cook for another 2 minutes. Lower the heat to medium and stir in the flour and the remaining 1 tablespoon olive oil. Add the 1/3 cup Marsala and stir until the mixture is smooth. Whisk in the chicken broth and simmer for 5 minutes. Add the meatballs to the sauce and simmer for an additional 5 minutes to let the flavors blend. Serve hot, garnished with grated pecorino.