Line a 1/4 sheet pan with parchment paper, making sure there is a bit of overhang on all sides. Grease the parchment with a little cooking spray or butter. Alternatively, skip the parchment and just grease your pan.
In a medium bowl, stir together flour, baking soda, and salt, and set aside.
In a glass measuring cup or a small bowl, combine the cold coffee and milk. Set aside.
Put the chopped chocolate into a microwave-safe bowl and heat until just melted, stirring every 20 seconds or so. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment (or in a large bowl using a hand mixer), cream butter and sugar.
Add egg and vanilla extract and mix until combined, scraping bowl as needed.
Add 1/3 of the flour mixture and mix until just combined. Then add 1/2 of the coffee/milk mixture and mix until combined, scraping bowl as needed. Repeat with another 1/3 of the flour mixture and the rest of the coffee/milk mixture. Scrape bowl.
Add the remaining 1/3 of the flour mixture and mix just until combined.
With the mixer running on low speed, slowly add the melted chocolate. Mix until fully combined, scraping the bottom of the bowl to make sure everything is mixed well.
Spread batter into the prepared pan, using a spatula to spread it evenly and into all 4 corners.
Bake for 18-22 minutes, or until a toothpick inserted into the center of the cake comes out clean or with just a few moist crumbs.
Move to a wire rack to cool completely (at least 2 hours) before frosting.
For the Frosting:
In the bowl of a stand mixer fitted with the whisk attachment, add the egg white, sugar, cream of tartar, and vanilla extract and mix until combined.
Add boiling water. Start beating on low speed, slowly increasing to medium-high speed(this helps keep the boiling water from splashing out of the bowl). Beat at medium-high speed until you have stiff peaks (about 7-9 minutes)
Spread evenly onto the cooled cake and sprinkle with coconut, if desired.
Note: this makes a lot of frosting. If you don't want to use all of it on your cake, set some aside in the freezer for future frosting needs. It freezes beautifully.
Stored in the refrigerator, the cake will keep for up to a week. For best texture, allow to come to room temperature before serving.