Combine the sugar, brown sugar, flour and cinnamon in a medium bowl. Using your hands, mix the cold butter into the mixture until it resembles coarse sand. Place in freezer while you make the doughnut batter.
Make the Coffee Cake Doughnuts:
Preheat the oven to 425°F. Spray two doughnut pans with cooking spray.
In a large bowl, beat together the butter, vegetable oil, granulated sugar and brown sugar until smooth, about 2-3 minutes.
Add egg and vanilla and beat to combine.
Whisk the flour, baking powder, cinnamon, baking soda and salt in a separate bowl.
Stir half of the dry ingredients into the wet ingredients until just combined. Do not overmix. Stir in the milk. Add the remaining dry ingredients and stir until no streaks of flour remain. The batter will be fairly thick.
Spoon the mixture into a large ziplock bag or piping bag. Snip off the tip and pipe the mixture into the prepared doughnut pans, filling to about 1/4″ from the rim.
Sprinkle with a generous layer of the frozen brown sugar crumble mixture, reserving approximately 1/3 of the mixture for after baking.
Bake for 11 minutes. Remove from oven and top with the remaining brown sugar crumble mixture. Let cool in pan for 5 minutes, then use a fork to gently loosen them from the pan and remove to a wire rack.
Drizzle with doughnut glaze and enjoy immediately.
Make the doughnut glaze:
Mix the powdered sugar and milk in a small bowl until combined. Add more milk as needed to get the desired consistency.