Freeze the ice cream layer first: Line the base of a 9-inch springform pan with parchment and, if you have it, add a parchment or acetate collar around the sides. Let the ice cream sit at room temperature for 10–15 minutes until scoopable, then pack it evenly into the pan and smooth the top. Freeze until very firm, 1–2 hours.
Heat oven to 325°F (165°C). Lightly grease two 9-inch cake pans and line the bottoms with parchment rounds.
Whisk the dry mix: In a bowl, combine flour, baking soda, salt, cornstarch, and espresso powder.
Make the wet base: In a large bowl, whisk the melted, cooled butter with the brown sugar and granulated sugar for 1–2 minutes until thick and slightly lightened. Whisk in the molasses, egg, egg yolk, and vanilla until glossy and fully blended.
Combine: Add the dry ingredients to the wet and fold just until no dry streaks remain. Stir in the chocolate chips. Cover and chill the dough for 10 minutes to make it easier to handle.
Bake the cookie rounds: Divide dough between the two pans and press into even layers (a flat-bottomed measuring cup helps). Dock each round a few times with a fork to prevent bubbling. Bake 14–16 minutes, until edges are set and centers are matte but still soft. Cool 15 minutes, then turn out to a rack and cool completely. For tidy assembly, freeze the cooled cookies for 20 minutes.
Assemble in the springform: Remove the firm ice cream disk from the freezer. Lift it out using the parchment and set aside. Place one cookie round into the springform as the base. Lay the ice cream disk on top, peel away the parchment, then top with the second cookie round. Press gently to level. Freeze until solid, 6–8 hours or overnight.
Finish the cake: Release the springform and remove the collar. Scatter the mini chocolate chips onto a rimmed sheet. Roll the cake’s ice cream edge in the chips to coat, or press chips on by hand. If the ice cream softens, pop the cake back into the freezer for 10 minutes and continue.
Serve and store: Let the cake stand at room temperature 5–8 minutes to soften slightly before slicing. Return leftovers to the freezer well wrapped; best within 5–7 days.