Preheat the oven to 350°F. Grease and flour three 8-inch round cake pans, and line with parchment circles.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, cocoa powder, baking soda, baking powder, and salt. Mix on low speed until blended.
In a heatproof bowl, slowly melt chocolate chips either in the microwave or with a double boiler. With either method, stir with a rubber spatula often to prevent burning. Once melted, set aside.
In a medium bowl, whisk together the milk, sour cream, vegetable oil, eggs, and vanilla. Pour the mixture into the bowl with the dry ingredients, then add the melted chocolate. Mix on low speed until the mixture comes together, then turn up to medium speed and beat for about 30 seconds. Stop the mixer and scrape down the side and bottom of the bowl with a rubber spatula, then beat again on medium speed for an additional 10 seconds.
Divide the batter evenly among the prepared cake pans and bake for 25 to 30 minutes, until a toothpick inserted in the center of the cakes comes out clean. Allow the cakes to cool in the pans for 20 minutes before turning them out onto a wire rack. Remove the parchment paper from the bottom of each cake, turn them right side up, and leave them to cool completely.
Cake layers will keep at cool room temperature, wrapped tightly in plastic wrap, for up to 24 hours.