Position an oven rack in the middle of the oven and preheat the oven to 350F [180C]. Grease an 8 in [20 cm] baking pan and line the bottom with parchment paper.
In a large bowl, whisk together the granulated and brown sugars, cocoa powder, oil, crème fraîche, egg, yolks, baking soda, baking powder, vanilla, and salt until well combined. Add the flour and use a spatula to stir until just combined. Slowly pour the coffee into the batter and mix until just combined.
Pour the batter into the prepared pan. Tap the pan on the counter a few times to remove any air bubbles.
Bake for 24 to 30 minutes, rotating the pan halfway through, until a wooden skewer or toothpick inserted into the center comes out with the tiniest bit of crumb. Transfer the pan to a wire rack and let cool for 20 minutes. Turn the cake out onto the rack, remove the parchment paper, and let cool completely. Top the cake evenly with the buttercream, the slice and serve. Store the cake in the refrigerator, covered, for up to 2 days.
For the buttercream
In the bowl of a stand mixer fitted with a paddle, beat the butter on medium speed for 2 to 3 minutes, until creamy. Add the cocoa powder, corn syrup, vanilla, and salt and mix again on medium speed for 2 to 3 minutes, until light and creamy. Scrape down the sides of the bowl, add the crème fraîche, and mix on low speed until combined. Add the confectioners' sugar and mix on low speed until combined, then add the tablespoon of water and mix again on low for 1 to 2 minutes, until combined and creamy The more you mix, the lighter the frosting will get. Finish mixing with a spatula, making sure all the ingredients are evenly incorporated.