Combine the lemon, parsley, garlic, oil and salt in a blender until mostly smooth.
Toss the chicken with the gremolata and marinate overnight or for at least 4 hours.
Preheat the oven to 350°F.
Remove the chicken from the marinade, discard marinade and season chicken with salt and pepper. Roast on a baking sheet until cooked through, about 22 minutes.
Heat a large skillet over medium heat with the olive oil, garlic, and rosemary until the garlic is toasted. Add the onion, pepper, tomato sauce, wine and chicken broth and bring to a boil. Reduce to a simmer and cook for 15 minutes.
Place the roasted chicken in a braising dish with a lid. Pour in the sauce, cover tightly and braise for 45 minutes, until the meat is very tender.
In a large pot, bring the water, milk, butter and salt to a boil. Reduce heat to low and whisk in the polenta. Cook, stirring regularly, for 20 minutes. Fold the cheese into the polenta and cook for 10 more minutes. Season with salt and pepper.
Toss the mushrooms with the gremolata ingredients on a sheet pan and roast for 15 minutes, until golden.
To serve, divide the polenta, chicken and mushrooms between plates. Top with parmesan and parsley.