Add ¼ cup of sugar along with the lemon juice to a heavy bottomed 3-quart saucepan.
Add the blackberries and blueberries to the saucepan and heat over medium/low heat, stirring frequently, until the berries breakdown and exude their juices (approximately 15 minutes). Remove from the heat and allow to cool for 5-10 minutes.
Place a fine mesh sieve over a bowl and pour the berries and their liquid into the strainer.
Use a spatula to push the berries into the strainer to release all the juice they can leaving the pulp and seeds in the strainer.
Dispose of the pulp and seeds and set the juice aside for the gelato. If making in advance, feel free to refrigerate the juice in an airtight container at this point until ready to move on to the next steps.
To make the gelato base:
In the same saucepan, add the whole milk, heavy cream, 1 and ⅔'s cup sugar, instant dry milk, reserved berry juice and vanilla.
Whisk well to combine and simmer the mixture (stirring frequently) over medium-low heat until the sugar fully dissolves (approximately 4-5 minutes) and the gelato is a lovely lavender color. Remove from the heat.
Allow the gelato base to cool for about 30 minutes and then transfer it into an airtight refrigerator container and refrigerate at least 8 hours, or up to 48 hours.
After the gelato has chilled pour it into the frozen drum of an ice cream machine and start the machine, churning until firm, about 2 hours.
Notes
1 serving = 1 cupStore and freeze the finished gelato in an airtight freezer container for up to 2 months.