Brown sugar icing:
1Preheat the oven to 350°F.
2In a medium bowl, stir together the flour, baking powder, spices, and salt. In a larger bowl, beat the butter, oil and sugar until light and fluffy. Beat in the eggs and vanilla.
3Add about 1/3 of the flour mixture to the butter mixture and stir it in by hand or with the electric mixer on low speed, just until it’s combined. Add about half the milk in the same manner, then another third of the flour, the rest of the milk, and the rest of the flour, mixing just until the batter is blended.
4Divide the batter between paper-lined muffin tins and bake for 25 minutes, or until the tops are springy to the touch. Let them cool tipped in their cups to allow steam to escape. Let cool completely before frosting.
5To make the frosting, In a small bowl or pot, stir together the brown sugar, water and vanilla. If you like, heat to a simmer to dissolve the sugar, or let it sit for a bit, stirring occasionally until the sugar is more or less dissolved. (It doesn't much matter if there's still some graininess in the bottom.) In a medium bowl, beat the butter until creamy; add the icing sugar and about 3/4 of the brown sugar syrup and beat until smooth. Add more sugar or syrup as needed until you have a soft, spreadable icing.