Mascarpone Polenta with Brown Butter Mushrooms

Name: Mascarpone Polenta with Brown Butter Mushrooms
Ingredients: 5 cups water; 1 cup cold unsalted butter (8 ounces), divided; 2 garlic cloves, smashed; 4 teaspoons kosher salt, divided, plus more to taste; 1 teaspoon black pepper, divided, plus more to taste; 1 cup uncooked coarse-ground polenta (such as Moretti); 1/3 cup mascarpone cheese (from 1 [8-ounce] container); 2 pounds mixed fresh wild mushrooms (such as oyster and chanterelle), torn into 1 1/2-inch pieces; 1/4 cup plus 2 tablespoons olive oil , divided; 3 tablespoons apple butter; 4 large eggs; 1/4 cup chopped mixed fresh herbs (such as chives, flat-leaf parsley, and tarragon), plus more for garnish