Set up the double boiler (bain-marie). Find a pot to hold a large metal bowl without touching the bottom. If you plan on whipping your eggs/yolks, use the stand mixer bowl for this step. If not, use another metal bowl.

Fill the pot with water so the bowl is directly above the water, but not touching it. The closer the bowl is to the surface, the better, as this is where heating will be most efficient. Remove the bowl for now and heat the water to a gentle simmer.

Add the eggs (or the yolks and water for the yolk-only substitution) to a large heatproof mixing bowl. Add the white granulated sugar, then immediately mix with a spatula until it is evenly distributed.

Place the bowl on top once your double boiler is at a gentle simmer. Use a spatula to gently stir periodically and heat until the egg syrup reaches 160°F (71°C). Then, pour the hot syrup into a heat-proof bowl or measuring cup.

Optionally, you can whip the egg syrup to create a runny foam after it reaches the desired temperature. However, there is little difference in the final buttercream when doing this. Place the bowl in your stand mixer and use the whisk attachment to mix on high speed until the mixture becomes slightly more voluminous and paler. Then, pour the egg foam into a heat-proof measuring cup.

Allow the egg syrup (or foam) to cool to room temperature before adding to the butter.

In a small bowl, add the Dutch-processed cocoa powder. Pour in your boiling water and stir until all the lumps disappear and you form a very thick paste. Allow the paste to cool completely before using.

Add the slightly softened butter to the stand mixer bowl. With the whisk attachment, mix on high speed for about 2 minutes, scraping down the sides at least once. After mixing, the butter will be lighter in color, smooth, soft, and will form a ball inside the whisk.

Add the powdered sugar, vanilla extract, and a small pinch of salt to the butter, then whisk again at medium-high speed for 1-2 more minutes.

Ensure your egg syrup (or foam) is at room temperature, then add 4 to 5 additions, whipping for about 30 seconds after each. Once all the egg syrup has been added, mix for an additional minute on high speed.

Add the cooled cocoa paste to the frosting and mix for an additional minute. Scrape down as needed to ensure all the cocoa paste is incorporated.

The whisk attachment introduced many air pockets, so switch over to the paddle attachment and mix on low speed for at least a few minutes to smooth out the frosting. Give the frosting a taste and adjust for vanilla and salt, if needed.