Combine flour, sugar, and salt in the basin of a food processor and pulse to combine.
Scatter butter pieces over the surface and pulse in 1-second increments until butter is cut into the flour mixture. Pea-sized butter pieces may remain.
Add sour cream and pulse again just until dough begins to cling together. Turn dough out onto a clean, lightly floured surface and work into a 1” thick disk. Wrap in plastic wrap or wax paper and refrigerate for at least 30-60 minutes. Shortly before you are ready to roll out your dough, prepare the filling and egg wash:
Stir together brown sugar, flour, cinnamon, and salt until combined.
Add melted butter and vanilla extract and stir until combined and mixture is paste-like. Set aside.
Prepare egg wash by whisking together egg and water until well-combined.
Preheat oven to 400F (205C) and line a baking sheet with parchment paper.
Transfer chilled pastry dough to a clean, lightly floured surface and use a lightly floured rolling pin to roll dough into a rectangle that’s between ⅛-¼” thick (0.5cm). Cut into approximately 2 ½”x4” (6x10cm) rectangles (you can use a sharp knife or pizza wheel, but I like to use the cookie cutter I linked to in the equipment section in the recipe card).
Place one cut-out rectangle on prepared baking sheet and lightly brush with an even coating of egg wash.
Scoop 1 ½ Tablespoons of filling and drop it into the center of the rectangle. Cover with another rectangle cut-out of pastry dough and use a fork to crimp the edges closed to seal. Brush the surface of the pop-tart with egg wash and use a toothpick or the tip of a sharp knife to poke 6-8 holes through the top layer of dough (to vent the pop-tarts).
Repeat with remaining dough until you have formed all of your pop-tarts. Space the pastries at least 2” apart on baking sheet, you’ll likely need to bake two trays, keep the second tray in the refrigerator while the first one bakes.
Transfer to 400F (205C) preheated oven and bake 15-17 minutes, until beginning to turn golden brown. Allow to cool on baking sheet before topping with icing.
Whisk together powdered sugar, brown sugar (break up any clumps in the brown sugar), 1 Tablespoon of milk, corn syrup, vanilla extract, and cinnamon. Add more milk as needed, a small splash at a time, until icing is smooth and the ribbon of icing that drizzles off the whisk holds its shape for a second or two before dissolving back into the bowl.
Drizzle icing over pop-tart and allow to harden before serving and enjoying.