Preheat your oven to 350°F (175°C). Line a large baking sheet with parchment paper.
In a large bowl, cream together the softened butter, white sugar, and brown sugar until the mixture is light and fluffy, about 3 minutes.
Add the egg and vanilla extract to the butter mixture, beating until fully combined.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt.
Gradually stir the dry ingredients into the wet ingredients until a stiff dough forms.
Scoop rounded tablespoons of dough onto the prepared baking sheet, spacing them 2 inches apart.
Bake for 10 to 12 minutes, or until the edges are set but the centers still look slightly soft.
Remove from the oven and let the cookies cool completely on the baking sheet for at least 30 minutes.
Once cool, turn half of the cookies upside down. Place a generous scoop of softened ice cream onto each, then top with the remaining cookies, pressing down gently until the ice cream reaches the edges.
Wrap each assembled sandwich individually in plastic wrap and freeze for at least 4 hours before serving.