Arrange an oven rack in the upper third of the oven and another in the lower third. Heat the oven to 350 degrees. Lightly coat the bottom and sides of 3 (8-inch) cake pans with butter, then line the bottoms with rounds of parchment paper and smooth the parchment to eliminate air bubbles.
In the bowl of a stand mixer fitted with the paddle attachment, combine the cake flour, granulated sugar, salt, baking soda and baking powder; mix on low speed just until combined.
Bloom the cocoa: In a medium bowl, whisk to combine the cocoa powder and hot coffee. Add the crème fraîche and vanilla, then whisk until completely smooth. Set aside.
In a separate medium bowl, whisk the 5 whole eggs and 2 yolks until smooth; set aside.
Mix the batter: To the stand mixer bowl, add the oil and half of the cocoa mixture; mix on low speed just until the flour is completely coated in the liquid ingredients. Pause the mixer and scrape down the sides and bottom of the bowl, then mix on medium until smooth and pasty, about 30 seconds. Pause and scrape down the sides and bottom of the bowl again, then add the whisked eggs and remaining cocoa mixture. Beat on medium speed, scraping down the sides and bottom of the bowl once or twice, until the batter is very silky and has the consistency of thin pancake batter, about 1 minute.
Pour the batter into the prepared pans, dividing evenly. (For uniform layers, use a scale – you’ll get about 550 grams of batter per pan.) Transfer the pans to the oven, positioning two on the upper rack and one on the lower, staggering them so the pans above don’t block the one below. Bake, rotating the top two pans side to side and rack to rack after 20 minutes, until the tops of the cakes are domed, springy to the touch and a toothpick or cake tester inserted into the centers comes out clean, 25 to 30 minutes.
Remove the pans from the oven and let cool completely on a wire rack.
To make the buttercream, start with the pudding base: Place the chocolate in the clean bowl of the stand mixer fitted with the paddle attachment. In a small saucepan, combine the milk and ¼ cup/50 grams of the sugar. Bring to a simmer over medium, stirring occasionally to dissolve the sugar, then remove from the heat and set aside. In a medium bowl, add the cocoa powder, flour, salt and remaining ¼ cup/50 grams sugar, and whisk until combined. Add the yolks to the bowl and whisk slowly to combine (it will seem very thick at first, but will loosen up), then whisk more vigorously until the mixture is pale, thick and smooth. Whisking the yolk mixture constantly, slowly stream in about half of the warm milk mixture, then whisk the contents of the bowl back into the saucepan. Set the saucepan over medium-low heat and cook, whisking constantly and scraping against the sides, until thick and bubbling, 1 to 3 minutes. Immediately scrape the pudding over the chocolate in the stand mixer bowl.
Make the buttercream: Let the pudding mixture sit for a few minutes, so the heat gently melts the chocolate, then mix with the paddle on low speed until the chocolate is completely melted and the pudding is smooth and glossy. Increase the speed to medium and beat until the outside of the bowl is cool to the touch, 10 to 15 minutes. With the mixer running, add the butter a couple of pieces at a time, waiting for each addition to incorporate into the pudding before adding the next, until you have a smooth, glossy, spreadable frosting. Stop the mixer, thoroughly scrape the sides and bottom of the bowl, and add the vanilla. Beat once more just until evenly mixed.