In a medium saucepan, stir the milk and sugar together over medium-high heat until just about to boil.
While the milk is on the stove, place a large bowl on the counter and pour the heavy cream inside it. Cover the cream and bowl with a fine mesh sieve or strainer, set aside.
In a medium bowl, whisk the egg yolks and vanilla extract. Carefully pour the warm milk into the egg mixture, whisking constantly. Then, pour the egg & milk mixture into the saucepan that you used for the milk.
Heat over medium heat, stirring constantly, until it thickens enough to coat the back of a spoon. Once thickened, strain the egg & milk base into the heavy cream. Once all liquid has been pushed through the strainer, stir in the blackberry puree and fresh lemon juice.
Cover bowl with plastic wrap and place in the refrigerator for 2-3 hours, until thoroughly chilled.
Remove blackberry mixture from the refrigerator and churn in your ice cream maker according to manufacturer's directions.