Mushroom Ragout with Soft Polenta

Name: Mushroom Ragout with Soft Polenta
Ingredients: 5 1/2 cups chicken stock or low-sodium broth; 3/4 cup whole milk; 1 cup fine polenta (not instant); Kosher salt; 6 tablespoons unsalted butter; 1/4 cup freshly grated Parmigiano-Reggiano cheese, plus more for serving; Freshly ground pepper; 1/4 cup extra-virgin olive oil; 1 1/2 pounds mixed mushrooms, such as cremini, chanterelles and stemmed shiitake, thickly sliced or halved; 1 1/2 tablespoons sherry vinegar; 1 tablespoon fresh lemon juice; 1 teaspoon finely chopped thyme; 1/4 teaspoon minced rosemary; 2 tablespoons chopped parsley, plus more for garnish