Preheat the oven to 425 degrees F (220 degrees C). Line a baking sheet with parchment paper.
Roll each sheet of puff pastry into a 12×9-inch rectangle. Place 1 puff pastry sheet on the prepared baking sheet and prick all over with a fork. Set the other one on top and refrigerate until ready to assemble.
For cookie dough, beat butter, brown sugar, white sugar, vanilla, and salt together with an electric mixer until light and fluffy, 1 to 2 minutes. Add egg and mix until combined. Add flour and chocolate chips and mix on low until a soft dough forms.
Remove puff pastry from the refrigerator; set the top layer aside. Spread cookie dough on the layer of puff pastry that was pricked with the fork, leaving a 1/2-inch border around the edges.
For egg wash, whisk egg and milk together in a small bowl, and brush the exposed edges of the pastry, then add remaining sheet of puff pastry on top; pinch the edges to seal. Crimp with a fork to make a border and poke a few vent holes on top of the pastry. Brush the top of the tart with egg wash.
Bake in the preheated oven until golden brown and puffed, 20 to 22 minutes. Let cool for 10 minutes.
For glaze, whisk confectioner’s sugar, brown sugar, warm milk together until smooth. Spread on top of the baked pop tart, again leaving a small border around the edges. Sprinkle the top with mini chocolate chips to finish.
Let the glaze set for 10 minutes, then cut the poptart into squares and enjoy!