Beat brown sugar, unsalted butter, and white sugar together in a bowl using an electric mixer until light and fluffy, 3 to 4 minutes. Mix in egg and vanilla extract until smooth. With the mixer on low speed, add flour, baking soda, and salt and blend until batter is just mixed. Fold chocolate chips into batter. Batter will be thick. Chill in the refrigerator at least 30 minutes to overnight.

Preheat the oven to 350°F (175°C). Line a 9-inch cake pan with parchment paper.

Spread batter into the prepared cake pan.

Bake in the preheated oven until edges are golden brown, about 20 minutes. Slightly cool cake in the pan before transferring to a wire rack to cool completely, about 30 minutes.

Beat shortening and butter together in a bowl using an electric mixer until smooth. Slowly beat in 3 cups of confectioners' sugar until smooth. Mix water and vanilla extract into confectioners' sugar mixture; add remaining confectioners' sugar and mix until buttercream is smooth. Add more water to thin, if needed.