Preheat oven to 350 degrees F. Generously spray a donut pan with nonstick cooking spray.
In a medium bowl, whisk together flour, cocoa powder, sugar, baking powder, baking soda, and salt.
In another medium bowl, whisk together yogurt, milk, egg, vanilla, and melted butter. Whisk in ¼ cup plus 2 tablespoons raspberry puree.
Pour the wet ingredients into the bowl with the dry and stir gently until just combined.
Scoop the batter into a quart-size zip-top bag: Fold the top opening of the bag down and over itself (to the outside) and set the bag in a cup or mug, so that you can scoop the batter into one corner of the bag. Scoop batter into the bag, remove it from the cup, and unfold the rim of the bag. Twist the top of the bag a few times so that it looks like a piping bag, and cut off the bottom corner of the bag. Pipe the batter into the six wells of the prepared donut pan.
Bake donuts for 9-10 minutes, until they are set and tops bounce back when gently pressed. Let cool in pan for 5 minutes, then transfer to a wire rack set over a rimmed baking sheet (to catch glaze drips) to cool for 10 minutes more.