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. (This keeps the temperature down.) Add salt, brown sugar, and milk, whisking until smooth. If it’s not cool yet, let it cool a bit more (you do not want the egg to scramble or the chips to melt) 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 1/2 cup (3 ounces or 85 grams) of the chocolate chips, stirring until flour disappears.
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, peeling off the parchment underneath. I like to then transfer my cookies to the freezer for 10 minutes so that they are fully, fully cold before we begin assembling the cake. If your ice cream is rock-hard in the freezer, transfer it to the fridge for these 10 minutes so it will be easier to scoop.
Assemble 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. This will make it easier to remove the cake. [An 8-inch springform can work too but risks leaking before the ice cream is fully frozen again; I’d still use some plastic inside for security.]
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. Use any plastic that has extended over the sides to cover the cookie, but it’s okay if it’s not fully covered.
Chill the ice cream sandwich cake for 3 to 6 hours and ideally overnight, until firm.
To finish and serve: 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 1/2 cup chocolate chips onto the sides of the cake. Cut the cake into 8 to 12 wedges and serve right away.
Do ahead: Ice cream cake keeps in the freezer for weeks, not that I’ve ever had one last that long.
Note
The (Abridged) Blondie Chipwich (individual ice cream sandwiches) from Smitten Kitchen Keepers: To make these as individual ice cream sandwiches: pour the batter into a single 9×13-inch pan. Once baked, cut the cookie in half, flip so that the top side of each cookie faces the outside, and fill with the ice cream, using the sides of the pan to help it keep its shape. Once frozen solid again, cut into 12 small squares, and press reserved chocolate chips onto sides. For the full recipe, and 99 others that (I’m biased but) are too fantastic to miss, grab a copy of Smitten Kitchen Keepers wherever you like to buy cookbooks.