Cinnamon Rolls with Icing

Name: Cinnamon Rolls with Icing
Ingredients: 2/3 cup whole milk; 2 (1/4-ounce) envelopes instant yeast (4 teaspoons total); 1/2 cup sugar; 2 large eggs, at room temperature; 3 cups unbleached bread flour; 1 teaspoon fine sea salt or table salt; 4 tablespoons (1/2 stick) unsalted butter, cut into 6 pieces and softened at room temperature; 8 tablespoons (1 stick) unsalted butter, cut into small pieces and softened at room temperature, plus additional butter for greasing the pan and plastic wrap; 3/4 cup packed light brown sugar; 1 tablespoon ground cinnamon; 2 teaspoons pure vanilla extract; 2 tablespoons unsalted butter; 1 cup confectioners' sugar, sifted after measuring; Instant-read thermometer, stand mixer fitted with a dough hook, 13- by 9-inch metal baking pan, parchment paper or foil