Preheat the oven to 350°F and place racks in the centre and bottom third of the oven. Line two baking sheets with parchment paper and set them aside.
In a stand mixer fitted with a paddle attachment, beat the brown sugar and butter on medium-high speed until light and fluffy, about 5 minutes. Reduce the speed to low and add the egg and vanilla. Beat until combined.
In a separate medium bowl, stir together the flour, baking powder, baking soda, and 1/2 teaspoon of the salt. Add the flour mixture to the butter mixture and beat until just barely combined. Add the popcorn and mix on low until the popcorn is fully incorporated. (At this point the batter can be stored in an airtight container in the refrigerator for up to 1 week.)
Drop large rounded spoonfuls of the batter onto the prepared baking sheets, leaving about I inch between them. Sprinkle the cookies evenly with the remaining 1/2 teaspoon salt. Bake for 12 to 15 minutes, rotating the baking sheets and switching their positions midway through the baking time, until the edges are golden brown and the centres are set. Remove from the oven and let cool on the baking sheet on a wire rack.
The cookies can be stored in an airtight container at room temperature for up to 3 days.