Combine blackberries, sugar, lemon juice, and water in a medium saucepan over medium-high heat.

Bring to a boil, then reduce to a simmer. Cook for 10-15 minutes, mashing the berries against the side of the pot as they soften, until broken down and syrupy.

For the smoothest texture, carefully pulse the hot mixture in a blender a few times.

Pour the mixture through a fine-mesh sieve set over a bowl. Firmly press the solids with a spatula to extract all the smooth purée. Discard the leftover seeds.

Stir the vanilla extract into the smooth sauce. (If a thicker consistency is desired, return to heat and whisk in the optional cornstarch slurry for 2 minutes).

Cool completely and store in an airtight glass jar in the refrigerator for up to 10 days.