For the butterscotch icing
1 1/4 cups dark brown sugar
1 cup granulated sugar
1/3 cup corn syrup
8 tablespoons butter
3 cups heavy cream
1/4 teaspoon salt
1 tablespoon vanilla extract
3 tablespoons dark rum

For the cake
3 1/3 cups flour
1/4 teaspoon baking soda
1/4 teaspoon baking powder
1 teaspoon salt
1/2 teaspoon freshly grated nutmeg
18 tablespoons butter, at room temperature
2 1/4 cups very fine turbinado sugar or granulated brown sugar (such as Domino’s Brownulated Sugar) or very fine turbinado sugar)
4 large eggs
2 teaspoons vanilla extract
1/4 cup dark rum
1 cup milk, preferably whole