Add the blackberries, sugar, cornstarch, and lemon juice to a medium saucepan over medium-high heat.

Mash the berries as the mixture warms. Once it starts bubbling, reduce the heat to medium and stir often until the sauce thickens, about 2 to 4 minutes total.

Transfer the blackberry sauce to a bowl and chill for about 30 minutes, or until fully cooled.

In a separate bowl, whisk together the sweetened condensed milk, vanilla extract, lemon extract, and half-and-half until smooth.

In another bowl, beat the heavy cream with a hand mixer or stand mixer until stiff peaks form, about 1 to 1 1/2 minutes.

Gently fold the condensed milk mixture into the whipped cream until the base is smooth and evenly combined.

Spread the ice cream mixture into an 8 x 8-inch pan or a similar freezer-safe container.

Spoon the cooled blackberry sauce over the top, then swirl it through the ice cream using a knife or fork.

Freeze uncovered for 3 to 5 hours, or until firm. For best texture, overnight freezing also works well.

Scoop the ice cream into chilled bowls or glasses and garnish with fresh blackberries or a sprig of mint.