Place blackberries and water in a medium saucepan over medium-high heat. Using a potato masher or a fork, crush the berries. Bring the berries to a simmer, then lower temperature to medium-low and simmer for about 2 minutes.
Remove pan from heat and pour berries into a fine-mesh strainer set over a glass measuring cup. Using the back of a ladle or spoon, press on the berries to squeeze out as much juice as possible. Discard solids. You should end up with about 2 cups of juice. Set aside to cool for 15 minutes.
In a medium saucepan, whisk the sugar, cornstarch, and salt together. Add the egg yolks, lemon juice, and zest, whisking until combined. Pour in the blackberry juice, and whisk until smooth.
Cook mixture over medium heat, whisking constantly until the filling has completely thickened and large bubbles begin to form on the surface, about 7-8 minutes.
Remove saucepan from the heat and immediately pour through a fine-mesh strainer set over a bowl, using the spatula to push all the filling through the strainer while leaving behind any bits of cooked egg. Whisk in the butter until fully combined.
Pour the filling into the prepared pie crust and smooth out the top. Allow to cool to room temperature before chilling in the refrigerator for at least 3 hours or overnight.