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For the cupcakes:
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Preheat oven to 350u00b0F. Line 2 cupcake pans (this yields 18-19 cupcakes, so it will be more like 1 1/2 cupcake pans) with cupcake liners and set aside.
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In a medium-sized bowl, sift together flour, cocoa powder, espresso powder, baking powder, baking soda, and salt.
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In the bowl of an electric mixer, beat butter until light, about 1 minute. With the mixer on medium-high speed, gradually add sugar; mixing until fluffy and very light in color, about 3 minutes, scraping down sides of bowl a few times. Add eggs and egg white one at a time, beating until almost completely combined after each addition. Add vanilla and beat briefly, about 10 seconds.
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Stop mixer and in a liquid measuring cup (or bowl), mix together buttermilk and coffee. Turn mixer on low speed and add flour mixture to butter/sugar mixture alternately with the buttermilk/coffee in 3 parts. Mix until almost completely combined; do not over-mix! Be sure to scrape sides of bowl with a rubber spatula a few times during alternations. Finish mixing by stirring batter together with rubber spatula until just combined.
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Divide batter evenly between cupcake liners (I used about 1/4 cup batter per liner). Bake for 14-17 minutes (1 pan baked longer than the other) or until cakes spring back when pressed lightly in the center and/or a toothpick or cake tester comes out with moist crumbs. Rotate pans halfway through baking.
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Transfer pans to wire rack to cool. Cool completely before frosting. While cupcakes are cooling, start frosting.
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For the dark brown sugar (and brown butter!) frosting:
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In a medium-small sized saucepan placed over medium-low heat, melt half of tbe butter until butter goes from lemon-like yellow color to a toasty brown, swirling the pan to ensure even cooking.
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Once we have some delicious brown butter, maintain the medium-low heat and to the saucepan with the butter, add brown sugar, cream, and salt. Cook until mixture begins to simmer, stirring occasionally. Remove from heat.
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Add eggs to mixture, stirring to combine. Return pan to heat and simmer, stirring constantly, for 2-9 minutes, or until mixture is thick and foamy. Cool completely (this took 1 1/2 hours).
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About 30 minutes before continuing, set remaining butter out to warm to room temperature. Once brown butter mixture is cool, transfer to the bowl of a stand mixer (a hand mixer can also be used). Cut room temperature butter into pieces and add to brown sugar mixture. Beat until butter is completely blended and frosting has lightened in color (the texture will also be smooth), about 1-2 minutes, occasionally scraping bottom and sides of bowl.
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Pipe or spread onto cupcakes and enjoy immediately or place in fridge until desired serving time!
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Notes:
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1. If you don't wish to add the coffee, just substitute with an additional 1/2 cup buttermilk.
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2. I had a little trouble with runniness in the frosting so here is what I did, in case you have the same problem: Add some powdered sugar (a little at a time, I added 2 1/3 cup total), switch to the whisk attachment on your mixer and whip! If still a bit runny, pop bowl in the freezer for a good 15 minutes-give a few stirs when you remove from freezer (and maybe one or two during the chilling process).
