Preheat the oven to 350ºF (177ºC). Note that as soon as you put the cake in the oven, you'll reduce the heat to 325ºF (170ºC). Grease a 9x3-inch (23x8-cm) springform pan and place a piece of parchment across the bottom before securing the sides, so it lines just the bottom of the pan. See photo.
To a medium, heat-safe bowl add the butter, cocoa powder and hot espresso. Stir to coat. It won't be enough liquid to really do much, but nudge it around. Add the chocolate and set over a medium saucepan of simmering water. Do not let the bowl touch the water or let the water come to a boil. Stir frequently until melted and smooth and set aside to cool.
In an electric stand mixer fitted with the whisk attachment, add the eggs, egg yolk, sugar, brown sugar, vanilla, espresso and salt and whisk on medium-high for about 8 to 10 minutes or until thick, light in color and doubled in volume.
Slowly add in the cooled, melted chocolate with the mixer on medium-low. Then run the mixer for 30 seconds on low to combine. Take the bowl out of the mixer and fold in the flour until just combined. Gently run a whisk through to make sure there aren't any flour pockets. Be gentle as you don't want to deflate the cake batter.
Pour into the prepared pan and place in the center of the oven. Immediately lower the temperature to 325℉ (170ºC) and bake for 55 minutes. Once done, set on a cooling rack. After 10 minutes run a knife along the edge of the cake and carefully remove the outer edge of the pan. Let cool completely before serving (if you can). It will deflate and crack a bit as it cools.I like the cake best the same day it is bake as the top will still be crispy. The next day it is still delicious, but the crispy top will soften.