Combine the blackberries, 100 g granulated sugar, and 20 g lemon juice in a medium heavy-bottomed saucepan. Cook over medium heat for 5 to 7 minutes, stirring and crushing about half the berries, until the juices simmer. Whisk the cornstarch with the cold water, stir it into the fruit, and boil for 1 full minute. The filling should look glossy, and a spatula drawn across the pan should leave a trail briefly. Transfer it to a shallow bowl, cover the surface, and refrigerate until completely cold.

Position a rack in the center of the oven and preheat to 175°C. Lightly grease three 8-inch round cake pans, line the bottoms with parchment rounds, and grease the parchment. Leave the pan sides only lightly coated so the batter can climb evenly. Set the pans on a large work surface and confirm they sit level. Preparing all three before mixing matters because the activated batter should be divided and baked promptly rather than waiting in the bowl.

Whisk the flour, baking powder, and 5 g salt in a medium bowl for at least 30 seconds to distribute the leavener. In a separate jug, whisk the room-temperature buttermilk, 45 g lemon juice, and 10 g vanilla until uniform. The dairy may look slightly thickened after the juice is added, which is expected. Keep both mixtures beside the mixer so they can be incorporated without delays or prolonged mixing once the flour begins hydrating.

Place 360 g granulated sugar and 12 g lemon zest in the stand mixer bowl and rub them together with your fingertips until the sugar feels damp and smells strongly of lemon. Add 170 g softened butter and beat with the paddle on medium speed for 3 to 4 minutes, scraping twice, until pale and fluffy. With the mixer running on medium-low, drizzle in the oil over 30 seconds. Continue mixing until the emulsion looks smooth, creamy, and free of visible oil.

Beat the eggs together briefly, then add them to the creamed mixture in four additions on medium-low speed, mixing until each portion disappears before adding the next. Scrape the bowl and paddle. On low speed, add the flour mixture in three portions, alternating with the buttermilk mixture in two portions, beginning and ending with flour. Stop when only a few flour streaks remain, then fold with a spatula until smooth. The finished batter should be thick, glossy, and evenly emulsified, not whipped or runny.

Divide the batter evenly among the prepared pans, using approximately 550 g per pan, and smooth the surfaces with a small offset spatula. Bake all three pans on the center rack for 24 to 29 minutes, rotating their positions after 18 minutes if the oven browns unevenly. The layers are done when their centers spring back, the edges begin pulling from the pans, and a tester shows moist crumbs but no wet batter. An optional center temperature is approximately 96 to 98°C.

Set the pans on wire racks and cool the cakes in their pans for 15 minutes. Run a thin spatula around each edge, invert onto the racks, remove the parchment, and turn the layers upright. Let them cool completely for about 90 minutes before trimming or frosting because residual warmth will loosen the cream cheese frosting. If the tops are domed, use a long serrated knife to remove only the raised portion, keeping each layer as level and equally thick as possible.

Beat 300 g softened butter with the paddle on medium speed for 2 minutes until smooth. Add the sifted confectioners’ sugar in four additions on low speed, followed by 2 g salt, 6 g lemon zest, 15 g lemon juice, and 5 g vanilla. Beat on medium for 2 minutes, then scrape thoroughly. Add the cold cream cheese one piece at a time on low speed, mixing only until incorporated. Stop when the frosting is smooth and holds a firm curved peak, since overbeating warms and loosens it.

Place one cake layer on an 8-inch cake board or serving plate. Fill a piping bag with about 250 g frosting and pipe a 1/2-inch-high dam around the outer edge. Spread half of the cold blackberry filling inside the dam without crossing it. Repeat with the second cake layer, another dam, and the remaining filling, then add the final layer cut-side down. Apply a thin frosting coat over the top and sides, trapping loose crumbs, and refrigerate the cake uncovered for 30 minutes.

Cover the chilled cake with the remaining frosting, using an offset spatula for the top and a bench scraper for the sides. Avoid repeatedly smoothing once the frosting warms; refrigerate it for 10 minutes if it becomes glossy or slack. Arrange the 125 g fresh blackberries around the top edge and scatter over the final teaspoon of lemon zest. Chill the decorated cake for at least 30 minutes to stabilize the layers. For clean slices, use a hot, dry chef’s knife and wipe it between cuts.