Place milk in the top of a double boiler and bring just to a simmer over gently boiling water over medium heat.
Meanwhile, whisk together eggs, egg yolks, and 3⁄4 cup of the sugar in a mixing bowl. Lightly whisk 1⁄4 cup of the hot milk into egg mixture, then whisk egg mixture into remaining milk in top of double boiler. Cook, stirring constantly with a wooden spoon, until mixture is thick enough to coat back of a spoon, about 15 minutes.
Transfer mixture to a mixing bowl; stir in 1 1⁄4 cups heavy cream; refrigerate until cold.
Combine blackberries, remaining sugar, and lemon juice in a mixing bowl. (Add more sugar if berries are tart.) Cover and refrigerate for 2 hours.
Stir into cream base and pour into an ice cream maker. Process according to manufacturer's directions. If desired, just before mixture is set, add 1 cup halved fresh blackberries.