In a mixing bowl, whisk together the flour, eggs, milk, sugar, salt, melted butter, and vanilla extract until smooth. Let the batter rest for 30 minutes.
Heat a non-stick skillet over medium heat and lightly grease it with butter. Pour about 1/4 cup of the batter into the skillet, swirling to coat the bottom evenly. Cook for 1-2 minutes until the edges lift and the bottom is lightly browned. Flip and cook for another minute. Repeat with the remaining batter, stacking the crepes on a plate.
In another bowl, beat the softened cream cheese, powdered sugar, heavy cream, lemon juice, and vanilla extract until smooth and creamy.
For the blackberry sauce, combine blackberries, sugar, lemon juice, cornstarch, and water in a saucepan over medium heat. Cook until the mixture starts to thicken and bubble, about 5-7 minutes. Remove from heat and let cool slightly.
To assemble, spread a generous amount of cream cheese filling on each crepe, fold, and top with the warm blackberry sauce.