Prepare the Cake Batter

1. Preheat oven to 325F. Brush the inside of a 9×13 baking pan with butter or oil.  Place a sheet of parchment paper over the bottom and up 2 sides of the pan.  Brush paper with butter or oil.
2. In a medium bowl, whisk together eggs, milk, oil and vanilla extract.  Set aside.
3. Into a large bowl, or the bowl of a stand mixer, sift the sugar, flour, cocoa, baking powder, soda and salt. Add the bowl of wet ingredients to the dry ingredients and mix on medium speed with the whisk attachment for 2 minutes in the stand mixer or with a 4-5 minutes with a hand mixer. While continuing to mix, add the boiling water and mix until well-combined, about 30 seconds.
4. Pour the batter into the prepared baking pan.  Bake in preheated oven 30-35 minutes, until a skewer inserted into the center comes out clean.  The cake should be beginning to pull away from the edges as well.  Cool on a rack for at least 30 minutes before removing.  Allow to cool completely prior to brushing with syrup and frosting.

Make the Vanilla Syrup

1. In a small saucepan, mix together the vanilla, sugar and water.  Stir.  Heat over medium heat until mixture boils.  Continue to cook, stirring occasionally, for 2-3 minutes, until mixture becomes syrupy.  Remove from heat, pour into a small bowl, and let cool.

Prepare the Chocolate Frosting

1. In a double boiler, or in a metal or glass bowl set atop a pan of simmering water, gently heat the chopped chocolate, stirring occasionally, until melted and smooth. Be careful not to let the bottom of the bowl touch the hot water, or the chocolate could burn.  Remove from heat and let cool to room temperature.
2. In a large bowl or the bowl of a stand mixer and using the paddle attachment, whip together the butter and confectioners’ sugar.  It’s best to start at low speed, then, once they mix together, move up to medium speed.  This will prevent a mess of confectioners’ sugar coming from the mixer into your kitchen.  Mix the two ingredients until they are light and fluffy, which should take 6-8 minutes.
3. Slowly add the melted chocolate, which should be at room temperature, until incorporated.  Then add the vanilla. Continue mixing (Medium speed) 2-3 more minutes, until the frosting is spreadable.  This frosting can be a bit sensitive to temperature, so unless it is super-hot in your kitchen, it is best to leave it at room temperature prior to frosting the cake.  However,  if it begins to lose its fluffiness, you will need to refrigerate a few minutes so it can be spread easily.  If you prepare it in advance and refigerate it, you will need to bring it back to room temperature, then whip it up a bit with a mixer.  I’ve found that if it is refrigerated, I can warm it slightly on my microwave power setting of 4 (of 10) for 30-45 seconds, and then stir it and it will be smooth.  If this doesn’t work for you, you will likely need to whip it again.

Assemble the Cake

1. If you would like to frost the top and the sides of the cake, you will want to dump it out onto a tray or board to frost. Place the board or tray atop the cake pan, hold tight and flip quickly.  The cake should come out easily if you oiled both the pan and the parchment.  If not, use a knife or rubber spatula to be sure the cake is separated nicely from the pan edges.
2. Brush a little of the vanilla syrup (you may need only 1/2-3/4 of the syrup) over the top of the cake.  Then spread the frosting in a thick layer up the sides and over the top of the cake.  It is likely you will have a cup or so of the frosting leftover, possibly more if you frost the cake in the pan.