Add blackberries, ¼ cup sugar and lemon juice to a saucepan and cook over medium heat, until the berries have softened, about 15 minutes. Mash the berries as they cook to release all their juices.
Pour the blackberries through a wire mesh sieve mashing down with the back of a spoon, and, reserving all of the liquid. Discard the pulp. Let it cool while you make the custard.
Add milk, cream, ⅔ cups of sugar and salt to a saucepan over medium-low heat. Bring to a simmer and remove from the heat.
Whisk the egg yolks together in a large measuring cup or bowl. Gradually add 1 tablespoon of the hot milk mixture to the egg yolks, whisking until combined. Continue slowly adding the warm milk until the egg mixture is warm. Then slowly add the warm eggs to the hot milk, whisking constantly until incorporated.
Return to the heat and cook for about 5 minutes until it is slightly thick and coats the back of a spoon. (an instant thermometer will read 170º) Pour it through a wire mesh sieve into a bowl. Add the blackberry syrup and vanilla and let it come to room temperature, for about 1 hour. Refrigerate for at least 4 hours, or overnight.
Process according to the manufacturer’s directions. It should take about 20 minutes to become soft-serve ice cream. Pour it into a 9 x 5 x 3 loaf pan and cover with plastic wrap until firm or place in an ice cream storage container.