Preheat oven to 325 F.
In a small saucepan, heat the milk and cocoa powder on medium heat until the powder is fully dissolved. Do not let the milk come to a boil. Stir continuously, while breaking up the clumps of powder. Once dissolved, set aside to cool.
In a large mixing bowl, sift together sugar, baking soda, the first portion of cream of tartar, salt, and flour.
Create a well in the center of the dry ingredients. Inside it, add yolks, oil, vanilla extract, and chocolate milk mixture. Stir to combine until fully incorporated, do not overmix. Set the chocolate-flour mixture aside.
In the bowl of a stand mixer, using the whisk attachment, whisk egg whites on low speed until frothy, then add the second portion of cream of tartar.
Increase speed to medium, once whites have become opaque and bubble size has tightened up, add sugar in a slow stream.
Increase high speed, whip until stiff peaks form. Now you have a meringue on your hands!
Gently stir 1/3 of the meringue into the chocolate mixture to loosen it up.
Gently fold the rest of the meringue into the chocolate-flour mixture.
Remove the prepared shamrock pan from the freezer and pour the chocolate cake batter into the pan, starting at the center of the pan. Spread the batter out in an even layer throughout the pan, using an angled spatula.
Tap the pan on the counter to release large air pockets. Bake for 14-16 minutes until the top of the cake springs back when gently prodded in the center.
Let cool 5 minutes in the pan before removing. Invert cake onto a fresh piece of parchment on a wire rack and gently remove the original parchment paper covering the shamrocks. Put a fresh piece of parchment paper on the top, then flip again so that the top (shamrock) side is facing down.
Roll the cake gently, starting at the short end, layering the parchment paper between the rolls to prevent any sticking. It's important to do this while the cake is still warm, to avoid cracking.While the cake cools, make the filling.