Heat oven to 350°F (175°C) degrees and line two 8-inch round cake pans with rounds of parchment paper. Coat the exposed sides with butter or nonstick spray.

In a large bowl, melt butter half in the microwave or in a medium-sized pot on the stove, then stir until the butter finishes melting. Add salt, brown sugar, and milk, whisking until smooth.

Let it cool a bit more before adding the egg and vanilla, whisking until combined. Sprinkle batter with baking powder and mix very well into the batter. Stir in flour and ½ cup (3 ounces or 85 grams) of the chocolate chips.

Pour half the batter into each pan and smooth the top flat. Sprinkle the tops of each round with 2 tablespoons of the chocolate chips.

Bake for about 15 minutes, until cookies are golden at the edges and set but soft in the center. Transfer to a cooling rack for 5 minutes, then remove from the pans.

Transfer cookies to the freezer for 10 minutes so that they are fully cold before assembling the cake.

Line one of the 8-inch round cake pans with plastic wrap that extends up the sides of the pan in two directions.

Place first chocolate chip cookie round top side down in the bottom of the pan. Scoop vanilla ice cream all over and use a spatula to spread it evenly.

Place second chocolate chip cookie top side up on top and press it firmly against the ice cream. Chill the ice cream sandwich cake for 3 to 6 hours and ideally overnight, until firm.

Remove the cake from the freezer. Run a knife along the edges of the cake pan and then use the plastic wrap to help lift the cake out of the pan. Pat remaining ½ cup chocolate chips onto the sides of the cake.

Cut the cake into 8 to 12 wedges and serve right away.