Preheat the oven to 350℉ (180℃) and liberally grease a bundt pan. I recommend using a baking spray or a combination of shortening + flour.
In a mixer fitted with the paddle attachment, beat the butter and oil together until slightly combined (appr. 30 seconds). Add the sugar and continue beating on medium-high speed for 2-3 minutes, until very light and fluffy. Make sure to scrape down the sides of the bowl halfway through.
Add the eggs, one at a time, followed by the vanilla. Beat each egg in fully for around 20 seconds before adding the next. Once again make sure you’re occasionally scraping down the sides of the bowl.
In a separate bowl, mix the flour, pudding mix, baking powder, baking soda, and salt together until very well combined.
Whisk the milk and sour cream together in a small cup.
While running your mixer on the lowest speed possible, alternate adding ⅓ of the flour, half the milk, another ⅓ of the flour, the remaining milk, and then the remaining flour. Mix until the ingredients are just combined.
Transfer the batter to the bundt pan. Bake for 50-55 minutes, or until a toothpick comes out clean. Let the cake cool for 10 minutes in the pan before flipping over onto a rack to cool completely.
To make the icing, whisk all ingredients together until smooth. Pour over the cooled bundt cake and enjoy!