Whisk flour, baking soda, and salt together in a medium bowl.

In a large bowl, beat softened butter with both sugars on medium speed until light and fluffy, about 2 minutes. Scrape the bowl.

Beat in the egg and vanilla until combined, about 30 seconds.

Gradually add the dry ingredients, mixing on low just until no streaks of flour remain. Fold in chocolate chips.

Divide dough in half, place each half on a sheet of plastic wrap, shape into a 6-inch disc, wrap tightly, and refrigerate for at least 2 hours (or up to 48 hours).

Preheat oven to 350°F (177°C). Line two baking sheets with parchment.

Roll one disc of dough between two sheets of parchment to 1/4-inch thickness. Use a 2-inch round cutter to stamp out rounds. Transfer rounds to baking sheets, spacing 1 inch apart.

Bake 8-10 minutes, until edges are set and centers look slightly underdone. Let cool on the pan 5 minutes, then transfer to a wire rack to cool completely.

Repeat with second disc. Reroll scraps once. You should have 24 cookies total.

Line a 9x13-inch baking dish with plastic wrap, leaving overhang. Spread softened ice cream evenly into the dish, about 3/4 inch thick. Freeze until firm, at least 4 hours.

Use the same 2-inch cutter to stamp out ice cream rounds. Working quickly, sandwich one ice cream round between two cookies. Gently press to adhere.

Wrap each sandwich individually in plastic wrap and freeze for at least 1 hour before serving. Let sit at room temperature 2 minutes before eating.