Preheat your oven to 375°F. Remove the puff pastry sheets from the box and let them sit at room temperature for about 30 minutes, until pliable but still cold.

In a medium saucepan, combine the blackberries, granulated sugar, Meyer lemon juice, and 3 tablespoons of water. Bring to a boil over medium-high heat, then reduce to a simmer. Cook for about 3 minutes, stirring frequently and gently mashing the berries with a fork.

In a small bowl, whisk the remaining 3 tablespoons of water with the 1 tablespoon of flour until smooth. Pour this slurry into the berry mixture and continue cooking, stirring constantly, until the filling thickens—about 3 more minutes. Remove from heat and let cool completely; it will thicken further as it cools.

Lightly dust a clean work surface with some of the 1/2 cup flour. Unfold one pastry sheet onto the floured surface. Use your fingertip to gently press along the fold lines in a circular motion to smooth them out. Lightly flour the top of the pastry and roll it out into a 12-inch square.

Using a sharp knife, cut the pastry sheet into 4 equal squares (about 6 inches each). Repeat with the second sheet.

Place 1 to 2 tablespoons of the cooled blackberry filling in the center of each pastry square. Be careful not to overfill.

Fold one corner of the square over the filling to form a triangle. Press the edges together, then use a fork to crimp and seal the edges. Place the turnovers on a parchment-lined baking sheet.

Bake the turnovers for 20 minutes, or until golden brown and puffed. Let cool on the baking sheet for 5 minutes before transferring to a wire rack.

In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. For easy drizzling, pour the icing into a small zip-top bag, snip off a tiny corner, and drizzle over the warm turnovers. Serve warm or at room temperature.