Preheat oven to 400°F. Lightly grease a donut pan with butter and set aside.
In a small saucepan, heat butter, oil, and milk on low-medium heat until the butter has just melted. Set aside to cool to room temperature. When you dip your finger into it, it should feel neither warm nor cold.
Meanwhile, in a separate bowl, stir together flour, sugar, pistachios, baking powder, baking soda, and salt.
Once the liquids have cooled, whisk in the egg and vanilla extract until combined.
Pour the liquid mixture into the flour mixture and fold with a silicone spatula until just combined. Don't overmix!
Transfer batter into a piping bag and snip about an inch off the tip of the bag. Pipe rings of batter into the donut moulds, filling them about ¾ full.
Bake for about 10 minutes or until lightly browned and a toothpick inserted in the middle comes out clean. Don't overbake!
Remove pan from oven and let sit for 2-3 minutes before transferring donuts to a wire rack to cool completely.