Beat 1 1/2 cup butter and 3 cups sugar with an electric mixer in a large bowl until light and fluffy. The mixture should be noticeably lighter in color.
Beat eggs one at a time into butter mixture, allowing each egg to blend completely before adding the next. Beat in sour cream and 1 teaspoon vanilla extract with the last egg.
Gradually stir flour mixture into egg mixture, mixing until just incorporated.
Fold in 7 ounces flaked coconut, mixing just enough to evenly combine.
Pour the batter into prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 90 minutes. Cool in the pan for 10 minutes before removing to cool completely on a wire rack.
Whisk milk and cornstarch in a saucepan over medium heat until mixture becomes very thick, stirring constantly, about 5 minutes; set aside to cool completely.
Beat 1 cup sugar, 1/2 cup butter, and shortening with an electric mixer in a bowl until mixture is smooth and creamy.
Beat cooled milk paste and 2 teaspoons vanilla extract into sugar mixture to achieve a light and creamy frosting.
Spread frosting over completely cooled cake and sprinkle remaining 3 1/2 ounces flaked coconut over the frosting.
