For the cake:
Heat the oven to 325°F and arrange a rack in the middle. Coat the sides and bottoms of 2 (9-inch) cake pans with butter. Cut 2 rounds of parchment paper and place in the bottoms of the pans. Butter the tops of the parchment and dust the pans with flour, tapping out the excess; set aside.
Place the water in a small saucepan and bring to a boil over high heat. Remove from the heat, add the instant coffee, and stir until dissolved. Place the cocoa powder in a medium heatproof bowl and whisk in the hot coffee until there are no lumps. Let cool slightly, about 15 to 20 minutes.
Sift the measured flour, baking powder, baking soda, 1/2 teaspoon of the salt, and 1/2 cup plus 2 tablespoons of the sugar into a large bowl; set aside. Place the egg yolks, oil, and vanilla in a separate large bowl and whisk until combined. Pour the cooled cocoa mixture into the egg mixture and gently whisk until smooth. Pour the egg mixture into the flour mixture and whisk until the batter is smooth; set aside.
Place the egg whites, the remaining pinch of salt, and the cream of tartar in the bowl of a stand mixer and beat on medium-high speed until the egg whites begin to turn white and foamy, about 1 to 2 minutes. Slowly add the remaining 1/2 cup plus 2 tablespoons sugar, about 1 minute. Continue beating until stiff and glossy peaks form, about 2 minutes more.
Using a rubber spatula, stir about one-third of the egg whites into the batter to lighten the mixture. In two additions, gently fold in the remaining egg whites, making sure no white streaks remain but being careful not to deflate the whites. Equally divide the batter between the prepared pans. Bake side by side, rotating halfway through the baking time, until a toothpick inserted into the center comes out clean, about 40 to 45 minutes. Cool the pans on a rack for 10 minutes, then unmold the cakes onto the rack to cool completely. Wash the cake pans.
Using a serrated knife, trim a very thin layer from each cake top, just enough to even out the surface and expose the inside of the cake. Return the cakes to the cleaned cake pans; set aside.
For the tres leches:
Place the chocolate in a large, heatproof bowl. Place the half-and-half in a small saucepan over medium heat and bring it just to a boil. Pour the half-and-half over the chocolate and let it sit until the chocolate has softened, about 1 minute, then whisk until smooth. Whisk in the condensed milk, evaporated milk, rum, and salt until smooth and combined.
Divide the tres leches mixture into 2 equal portions. Starting with 1 cake and 1 portion, pour as much of the tres leches mixture over the cake as possible. Repeat with the second cake and tres leches portion. Set the cakes aside until they begin to absorb the mixture, about 20 minutes. Continue to add more of the mixture until all of it has been added, about 30 to 40 minutes more. Cover the cakes with plastic wrap and refrigerate 4 hours, or freeze 1 1/2 to 2 hours, until the cakes are easier to handle. (They may not completely absorb all of the mixture.)
Turn the cakes out of the pans by inverting each onto a plate over the sink, as some unabsorbed mixture may spill out.
For the filling and to assemble:
Place the cream in the bowl of a stand mixer fitted with a whisk attachment and whip until slightly thickened. Add the powdered sugar and vanilla and continue beating until thickened and smooth.
To assemble the cake, place 1 of the cakes on a serving platter. Using an offset spatula or long, thin knife, spread about one-third of the filling evenly on top. Place the second cake on top and evenly spread the remaining filling over the top and sides of the entire cake. Refrigerate for at least 15 minutes to let the whipped cream set.
Just before serving, garnish the cake: Hold the piece of chocolate with a paper towel in one hand, and use a vegetable peeler to shave the chocolate all over the top of the cake. Serve immediately.
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