Preheat oven to 375 degrees F (190 degrees C). Line 2 baking sheets with parchment paper.
Sift flour, 1 tablespoon sugar, and salt together into a bowl. Cut in chilled butter with a pastry blender or 2 knives until mixture resembles small peas. Add vanilla. Mix in cold water, 1 tablespoon at a time, until dough comes together and forms a ball.
Divide dough into 2 pieces and wrap with plastic wrap; chill until firm, at least 1 hour.
Beat softened butter, 1/4 cup sugar, and cinnamon together in a bowl to make filling.
Roll out dough on a lightly floured work surface to 1/2-inch thickness. Cut dough into rectangles with a knife. Drop 1 tablespoon of filling onto half of the rectangles. Cover with remaining rectangles. Press edges with a fork to seal. Poke a few holes in the top of each pastry with a toothpick.
Arrange pastries on the lined baking sheets.
Bake in the preheated oven until golden brown, 25 to 30 minutes. Cool pastries on a wire rack, about 15 minutes.