Preheat the oven to 350 degrees F. Grease a 6-cup donut pan with non-stick cooking spray. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, oil, granulated sugar and light brown sugar until smooth, about 3 minutes. Beat in the eggs followed by the vanilla extract.

In a large bowl, combine the flour, baking powder, baking soda, salt and lime zest. On low speed, mix in the flour in 3 additions alternating with the milk in 2 additions, starting and finishing with the flour

Transfer the batter to a piping bag or a Ziploc bag with a corner cut off, and pipe the batter into the pan. Bake for 10-12 minutes until a toothpick inserted into a donut comes out clean. Wait 3 minutes before carefully removing the donuts from the pan and letting them cool on a wire rack. Bake the remaining batter in the donut pan.

Once the donuts have cooled completely, make the icing. Puree the blackberries in the bowl of a food processor. Strain out the seeds and discard. In a large bowl, whisk together the blackberries, confectioners’ sugar, vanilla extract, lime juice and milk. Dip the tops of the donuts into the icing letting the excess drip back into the bowl. Let set for 1 hour before serving.