For the popcorn cookies: Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper.
Cream the butter and sugar with an electric mixer on medium speed in a large bowl until slightly lightened, 1 to 2 minutes. Scrape down the sides of the bowl and add the egg. Beat until well incorporated, about 30 seconds.
Whisk the flour, cornstarch, baking powder, kosher salt and 1 cup popcorn together in a separate bowl. Place the remaining popcorn in another bowl and set aside. Add the flour mixture to the creamed butter and mix on low speed until it just comes together to form a dough. Scoop a level tablespoonful of the dough at a time and roll into a ball. Dip one side of the ball into the ground popcorn and place, popcorn-side up, on the prepared baking sheets, about 2 inches apart. Slightly flatten each ball with your hand.
Bake until the cookies are slightly brown around the edges, 14 to 16 minutes. Allow the cookies to cool on the baking sheet for 5 minutes, and then remove to a rack to cool completely.
For the salted caramel: While the cookies cool, add the sugar to a medium saucepan over medium heat. Allow the sugar to melt completely around the edges before starting to stir or it will crystallize. When the sugar turns dark amber, after about 10 minutes, quickly turn off the heat and carefully whisk in the heavy cream. Allow the foam to subside, and then add the coarse salt. Allow the caramel to cool until it begins to thicken, about 5 minutes.
Spoon 1 tablespoon of caramel on the center of the bottom side of one cookie and carefully press closed with another cookie, making sure to push the caramel evenly to the edges.