Step 1

Place racks in upper and lower thirds of oven; preheat to 350°. Pulse butter, pretzels, potato chips, sugar, salt, and 1¾ cups (219 g) flour in a food processor until clumps form. Turn dough out onto a surface and knead gently to bring together.

Step 2

Roll out dough on a lightly floured sheet of parchment to just under ¼" thick. (If it feels very soft, cover and chill 20 minutes before proceeding.) Punch out rounds with lightly floured cutter. Transfer rounds to a parchment-lined baking sheet, spacing 1" apart. Gather dough scraps and repeat rolling and cutting process until no dough remains (you should get 36 rounds). Divide remaining cookies between 2 more parchment-lined baking sheets.

Step 3

Working in batches, bake cookies, rotating baking sheets top to bottom and front to back halfway through, until golden brown, 10–12 minutes. Let cool 5 minutes on baking sheet, then transfer half of the cookies to a wire rack. Let cool completely.

Step 4

Meanwhile, turn over cookies remaining on baking sheets and top each with a marshmallow, flat side down (make sure marshmallows stand up evenly or sandwiches will be crooked). If your oven has a convection fan, turn it off and bake cookies until marshmallows are softened and beginning to puff but not brown, about 4 minutes. Remove from oven and top with cooled cookies, letting weight of cookies gently push marshmallows to the edges. Let cool at least 1 hour.

Step 5

Melt chocolate and oil in a small heatproof bowl set over a small saucepan of barely simmering water (do not let bowl touch water), stirring occasionally, or melt in a small microwave-safe bowl in a microwave in 20-second bursts, stirring between bursts. Dip half of each sandwich cookie in chocolate, then sprinkle top of chocolate sides with pretzel salt. Chill cookies 15 minutes to harden chocolate before serving.Do ahead: Cookies can be made 2 weeks ahead. Store airtight at room temperature.