Farro Mafaldine with Black Truffle Butter and Mushrooms

Name: Farro Mafaldine with Black Truffle Butter and Mushrooms
Ingredients: 1 tablespoon unsalted butter, softened; 1 1/2 teaspoons black truffle paste or finely grated black truffle; 1/4 teaspoon fresh lemon juice; 1/2 teaspoon kosher salt, divided; 6 ounces mixed fresh wild mushrooms, stemmed and sliced (about 3 cups); 3 tablespoons extra-virgin olive oil, divided; 1/4 teaspoon black pepper; 8 ounces uncooked farro mafaldine pasta; 1 small garlic clove, thinly sliced; 1/2 cup vegetable broth (such as Zoup!); 2 tablespoons finely chopped fresh flat-leaf parsley; 1/2 ounce pecorino Sardo cheese, grated (about 2 tablespoon), divided