Gooey Butter Cake

Name: Gooey Butter Cake
Ingredients: ½ cup (1 stick) unsalted butter, room temperature, plus more for baking dish; ¼ cup warm whole milk (100 degrees); 1 envelope active dry yeast; ¼ cup granulated sugar; 1 teaspoon coarse salt; 2 large eggs; 2 cups all-purpose flour; 1 cup (2 sticks) unsalted butter, room temperature; 1 ⅓ cups granulated sugar; ¾ teaspoon coarse salt; 1 large egg, plus 1 large yolk; 1 tablespoon pure vanilla extract; 1 ⅓ cups all-purpose flour; ¼ cup light corn syrup; Confectioners' sugar, for dusting