Add the room temperature butter and the cocoa powder to a large mixing bowl and mix with an electric mixer on medium to medium-high until well mixed and smooth. Scrape down the sides of the bowl and mix again for another 20 seconds.
Add 3 cups of the confectioner's sugar plus 2 tablespoons milk and mix on low until combined, then on medium-high until fully incorporated.
Add the vanilla and the last 2 cups of confectioner's sugar. Mix on low until combined, then on medium-high until fully incorporated. Scrape down the sides of the bowl, then mix again for about 20 seconds.
Check the consistency. If it's too thick, add 1 tablespoon of milk and mix again. Repeat until the desired consistency has been reached.
For fluffy buttercream, mix for about 2 minutes on medium-high to high speed.
For smoother frosting, mix on low for a couple of minutes.
Timeline and How To Assemble:
Make sure the cake has cooled completely and that the raspberry filling is chilled. The cake does not have to be chilled, but it should not be warm.
Level the tops of the cake layers if needed however these cake layers should already cool flat on the top.
Add a layer of the cake to a cake plate, then add a thin layer of the chocolate frosting. This will keep the raspberry filling from soaking into the bottom cake layer and making it soggy. It's also good to pipe an icing dam just around the inside of the cake layer to keep the raspberry filling from squishing out later on.
Add the raspberry filling. Try not to add the filling close to the edges of the cake layer so that it doesn't squish out. Depending on how much the filling cooked down, there may be some extra left over. Don’t fill the cake too much. Either add the leftover filling to the top center of the cake after it's frosted, or add it to an airtight container, store in the refrigerator and use as a jam on toast, etc. Just make sure to use it within about 5 days or so
Next, add the last layer of cake, then cover the entire sides and top of the cake with the remaining chocolate frosting.
See the notes section for best ways to store this cake, how to serve it and for more questions.