Mississippi Mud Cake

Name: Mississippi Mud Cake
Ingredients: 2 large eggs; 1 cup whole or 2% milk; 1/2 cup sour cream; 1 cup plus 1 tablespoon vegetable oil, divided; 2 cups all-purpose flour; 1 cup packed light brown sugar; 1 cup granulated sugar; 1/2 cup Dutch-processed cocoa powder; 1 1/2 teaspoons baking powder; 1 teaspoon baking soda; 1 tablespoon vanilla extract; 1/2 teaspoon kosher salt; 1 (10-ounce) bag mini marshmallows (about 4 cups); 8 tablespoons (1 stick) unsalted butter; 1/3 cup whole or 2% milk; 1/4 cup Dutch-processed cocoa powder; 2 teaspoons vanilla extract; 1/4 teaspoon kosher salt; 2 1/2 cups powdered sugar