Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes.
Drain and transfer to a bowl. Mash potatoes using a potato masher or a fork.
Preheat oven to 400 degrees F (200 degrees C). Grease a baking sheet.
Heat oil in a skillet over medium heat; saute onions until golden brown, 10 to 15 minutes.
Mix potatoes, salt, and pepper into onions.
Remove skillet from heat and cool.
Roll each puff pastry sheet onto a work surface.
Cut each sheet into 4 squares.
Spread about 1/4 cup potato mixture onto each square in a strip down the middle; top with a hot dog. Cover each hot dog with about 1/4 cup more potato mixture.
Roll puff pastry around hot dog and filling; place, seam side-down, on the prepared baking sheet.
Brush puff pastry with egg.
Bake in the preheated oven until golden brown, 10 to 15 minutes.
