Beat together the chocolate cake mix, 1 cup milk, eggs, 1/2 cup melted butter, 1 teaspoon cinnamon, and 1 teaspoon vanilla extract in a bowl with an electric mixer on low speed until moist. Beat for 2 more minutes on medium speed.
Fill each muffin cup about 2/3 full of batter.
Sprinkle the cupcakes with cinnamon sugar.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 15 minutes.
Cool in the pans for 10 minutes before removing to cool completely on a wire rack.
Cream together 1/2 cup butter and shortening in a bowl until smooth.
Stir in sea salt, 1 teaspoon vanilla extract, 1 tablespoon cinnamon, cocoa powder, and 3 cups confectioners' sugar.
Stir in milk.
Mix in 2 more cups confectioners' sugar or as needed until desired consistency is achieved.