Coffee Cake Donuts

Name: Coffee Cake Donuts
Cuisine: American
Ingredients: ¼ cup unsalted butter, melted; ¼ cup light brown sugar; 1 cup all-purpose flour; ½ teaspoon ground cinnamon; 1 teaspoon baking powder; ½ teaspoon salt; ¼ cup granulated sugar; 1 egg, at room temperature; 1 tablespoon unsalted butter; ½ cup milk; 1 teaspoon vanilla extract; ½ cup confectioner’s sugar; ½ tablespoon milk