For the Blackberry Puree:

1. Place the berries, sugar, and lemon juice in a saucepan over medium heat. Stirring frequently, bring the mixture to a simmer. Cook for 10 minutes, or until the blackberries are mostly cooked down.
2. Remove the pan from the heat and pour the mixture through a fine-mesh strainer set over a bowl. Press the mixture through the strainer to remove any seeds. Discard the seeds.
3. Allow the puree to cool before using in the donut batter.

For the Donuts:

1. Preheat the oven to 350°F. Butter or spray your donut pan with nonstick cooking spray.
2. Whisk the flour, baking powder, baking soda, and salt together in a medium bowl. Set aside.
3. Whisk the egg, brown sugar, milk, blackberry puree, melted butter, and vanilla together until smooth and well combined.
4. Pour the wet ingredients into the flour mixture and stir gently until just combined, trying not to overmix the batter.
5. Spoon the batter into the prepared donut pan, filling each donut mold roughly ¾ full (alternatively, transfer the batter to the piping bag and pipe the batter into the pan).
6. Bake for 8-10 minutes, or until the batter is set, the donuts have puffed up, and the edges are lightly browned. Cool for a couple minutes in the pan and then carefully transfer the donuts to a wire rack to finish cooling.

For the Blackberry Glaze:

1. Place the blackberry puree in a small pot over medium heat. Bring to a simmer.
2. Whisk in the butter until it is completely smooth and melted.
3. Remove the pot from the heat and whisk in the powdered sugar, a little at a time, mixing until a smooth glaze forms.
4. If the glaze gets too thick as it cools, whisk in 1-2 teaspoons of water or lemon juice until it becomes a drizzling consistency.

To Glaze:

1. Dip each donut in the warm glaze and return them to the cooling rack to sit until the glaze has set, about 10 minutes.