Preheat oven to 425°F and line a 12-cup muffin pan with liners.
In a large bowl, mix together the butter, oil, and sugars until smooth.
Beat in eggs, one at a time. Add vanilla.
Add the baking powder, baking soda, pumpkin spice, and salt.
Using a rubber spatula to stir, alternate between stirring in the flour and buttermilk. Begin by gently stirring in 1 cup of flour. Add ½ cup of buttermilk. Add another 1 cup of flour, followed by the remaining ½ cup of buttermilk and stir just until combined. Finally, add the last ⅔ cups of flour until incorporated.
Divide the batter evenly between the 12 muffin cups. Bake for 15-17 minutes. Allow them to cool for a few minutes before removing them from the pan.
Cool for about 10 minutes on a rack before glazing.
Beat together all the ingredients for the glaze, then dip the top of each muffin in the glaze, allowing the extra glaze to drip back into the bowl. Let the first glaze layer harden before you dip them once more.