Step 1In the bowl of a stand mixture using the paddle attachment, or in a large bowl using a hand mixer, beat the butter, one-half cup sugar and the brown sugar at medium speed until light, 3 to 5 minutes. Add the egg and mix well to incorporate. Add the vanilla and mix until thoroughly combined.
Step 2In a medium bowl, sift together the salt, flour, cocoa powder and baking soda. Mix the dry ingredients into the butter mixture, one-half at a time, until thoroughly combined, scraping down the sides of the bowl as needed. Form the dough into a disc, wrap with plastic wrap and chill for 2 hours or longer.
Step 3Heat the oven to 350 degrees. Place the remaining one-third cup of sugar in a small bowl.
Step 4Shape the dough into 1-inch balls. If the dough starts to become sticky, return it to the refrigerator to chill briefly. Roll the cookies in the sugar. Lightly press down on the balls to flatten them to one-half inch. Place 5 to 6 small grains of sea salt on half the cookies (these will be the tops of the sandwiches), spacing the salt crystals out on top of the balls as much as possible.
Step 5Place the balls 2 inches apart on lightly buttered baking sheets. Bake until the surface puffs and begins to deflate, and the cookies darken slightly, 10 to 12 minutes. Rotate the pan halfway through for even baking. Cool the cookies for 1 minute on the baking sheet, then transfer the cookies from the sheet to a wire rack to cool.
Step 6To assemble, spread a generous one-fourth cup ice cream (slightly softened) over a bottom cookie, and carefully cover with a top cookie containing the salt crystals. Press firmly but gently to avoid breaking the cookies. Run a knife or offset spatula around the edges to clean up any ice cream hanging over the edge. Freeze for at least 1 hour to firm the ice cream. Repeat with all the cookies.