Position an oven rack in the center of the oven, and preheat to 175°C. Line the bottom of a 9-inch round cake pan with a parchment circle. Butter the parchment and the sides of the pan.
For the cookie layer: Whisk together the flour, baking soda and salt in a large bowl; set aside. Beat the butter and both sugars with an electric mixer on medium-high speed in a large bowl until light and fluffy, about 4 minutes. Add the egg, and beat until incorporated. Beat in the vanilla. Scrape down the bowl as needed. Reduce the speed to medium, add the flour mixture and beat until just incorporated. Stir in the chocolate chips.
Press the cookie dough into an even layer in the prepared cake pan, and bake until golden brown and a knife inserted in the center comes out clean, 25 to 30 minutes. Let the layer cool for about 10 minutes. It will collapse slightly as it cools. To make the layer even, press the bottom and sides down with the bottom of a measuring cup.
For the cake layer: Whisk together the flour, baking powder and salt in a medium bowl; set aside. Beat the granulated sugar and oil with an electric mixer on medium-high speed in a large bowl until light in color and fluffy, about 5 minutes. Beat in the eggs one at a time. Beat in the vanilla. Alternate folding the flour mixture and the milk into the batter with a spatula, adding the flour in three additions and the milk in two, starting and ending with the flour, until just smooth. Stir in the sprinkles. (The cake batter will be on the thin side, like a thick pancake batter.)
Pour the cake batter over the cookie layer, and bake until the middle of the cake springs back when pressed and a knife inserted in the center comes out clean, 1 hour to 1 hour 10 minutes. Run a knife around the edge of the cake, and let cool completely in the pan on a rack. Transfer the cake from the pan to a serving plate, with the cookie layer on the bottom.
For the frosting: Beat the cream cheese and granulated sugar with an electric mixer on medium-high speed in a large bowl until smooth and all the sugar has dissolved (rub some between your fingers; if it's gritty, keep beating). Add the cream and vanilla, and continue to beat until medium peaks form. Frost the cake, and decorate with more sprinkles.