Preheat the oven to 350°F (175°C).
In a mixing bowl, whisk together flour, baking soda, and salt.
In a separate bowl, cream the softened butter, brown sugar, and granulated sugar until light and fluffy.
Add the egg and vanilla extract to the butter mixture, mixing until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Scoop tablespoon-sized portions of dough onto baking sheets lined with parchment paper.
Bake for 10-12 minutes, or until the edges are golden and the centers are still soft.
Once cookies are completely cooled, scoop ice cream onto half of the cookies.
Top with the remaining cookies and gently press together.
Wrap each sandwich in plastic wrap and freeze until the ice cream hardens.