Preheat the oven to 350 degrees F and position the oven rack to the middle. Line two 24-cup mini-muffin tins with cupcake liners.
For the cookie crust: Combine the flour, salt and baking soda in a small bowl and set aside. Beat the butter and both sugars together in the bowl of a stand mixer fitted with a paddle attachment on medium-high speed until creamy, about 4 minutes (or in a large bowl if using an electric mixer). Add the egg and vanilla and beat until incorporated, about 1 minute. Adjust the speed to low, add the flour mixture and beat until incorporated. Fold in the chocolate chips.
Drop 1 heaping teaspoon of cookie dough into each liner and press down firmly. Bake until slightly puffed and golden, 12 to 15 minutes. Cool until warm to the touch, and then press down the crust once more. Cool completely.
For the brown sugar cake: Whisk the flour, baking powder and salt together in a medium bowl. In the bowl of a stand mixer fitted with the paddle attachment, beat the eggs and light brown sugar together on medium-high speed until light and foamy, about 2 minutes (or in a large bowl if using a hand-held electric mixer). While beating, gradually pour in the butter and then the vanilla.
While mixing slowly, add half the dry ingredients, then add all the milk, and follow with the rest of the dry ingredients. Take care not to overmix the batter. Drop 1 heaping teaspoon of batter on top of each cookie crust.
Bake until a tester inserted in the center of the cakes comes out clean, rotating the tins about halfway through, 10 to 12 minutes.
Cool the cupcakes on a rack in the tins for 10 minutes, and then remove from the tins and cool on the rack completely.
For the chocolate frosting: Combine the cocoa powder and butter in the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl if using a hand-held electric mixer). Mix on low until combined. Add the vanilla and salt. Continue to mix on low speed and add the confectioners' sugar and milk in alternating batches. Increase the speed to medium-high and mix until fully incorporated and smooth.
Fill a pastry bag fitted with a round or star tip and pipe small chocolate drops on top of each cupcake.