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To make the frosting: In a large heatproof bowl, combine the white chocolate, vanilla, and salt.
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In a small saucepan, heat the cream over medium-low heat until simmering.
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Meanwhile, in a large, heavy saucepan, combine the sugar, water, and cream of tartar. Place over medium heat and bring to a boil, whisking to dissolve the sugar. Continue to cook, swirling the pan occasionally (do not whisk), until the mixture turns uniformly deep amber in color (around 355°F on a candy thermometer). Immediately, remove the caramel from the heat and whisk in the cream until combined (be careful – the mixture will bubble and splutter at first).
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Pour the caramel mixture over the white chocolate and stir slowly with a heatproof spatula until the chocolate is melted and the mixture is smooth. Let sit for 5 minutes to cool slightly.
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Add the butter, one cube at a time, stirring slowly until incorporated. Transfer to an 8" square pan and refrigerate, uncovered, for 90 minutes, stirring gently every 30 minutes. The frosting will be ready to apply to the cake when it is the consistency of thick, creamy peanut butter and is firm enough to hold its shape when stirred or scooped. (If frosting is refrigerated for significantly more than 90 minutes, allow to come to room temperature for about 2 hours, or until soft enough to spread.)
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To make the cake: Preheat the oven to 350°F with a rack in the center. Grease two 8" round cake pans and line the bottoms with parchment; grease the parchment.
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In a medium bowl, whisk together the flour, baking powder, salt, and baking soda.
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In a large liquid measuring cup or another medium bowl, whisk the eggs, buttermilk, and vanilla to combine.
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In the bowl of a stand mixer fitted with the flat beater, beat the sugar and butter on medium-high speed until pale and fluffy, 4 to 5 minutes, pausing once to scrape the bowl.
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Reduce the mixer speed to low and add the flour mixture in three additions, alternating with the egg mixture in two additions, mixing until just combined after each addition and scraping the sides of the bowl between additions. Once the last of the flour is incorporated, increase the mixer speed to medium and mix until well combined, about 30 seconds more.
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Divide the batter evenly between the prepared pans (about 780g or 3 1/2 cups per pan) and smooth the tops; a small offset spatula is a helpful tool here.
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Bake for 45 to 50 minutes, until the cake layers are light golden brown and spring back when gently pressed; a toothpick inserted into the center should come out clean.
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Remove the cake layers from the oven and allow them to cool in the pans for 15 minutes. Loosen the edges with a small offset spatula or knife and gently invert the cakes onto a cooling rack. Peel off the parchment, turn the cakes right side up, and let cool completely.
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To make the soak: In a small bowl or liquid measuring cup, stir the milk, sugar, vanilla, and salt to combine.
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To assemble the cake: If the cake layers are domed in the middle, trim with a large serrated to level. Using the same knife, cut each cake layer in half horizontally to make four thin (approximately 1/2"-thick) layers.
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Place a small dollop of frosting in the center of a large serving plate, cake stand, or cardboard cake round to anchor the cake. Place one of the cake layers on top and brush with about a quarter of the soak.
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Scoop a generous 1/2 cup (about 200g) of frosting on top of the first cake layer and spread evenly, leaving about a 1/4" border. Repeat the process with the remaining cake layers.
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When all four layers are stacked, spread the sides of the cake with the remaining frosting.
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Storage information: Store the cake, covered, at room temperature for up to 5 days; press plastic wrap or parchment directly against any cut surfaces.
