Cake: Preheat oven to 350 degrees, with racks in upper and lower thirds. Butter five 8-inch round cake pans; line with parchment. Butter parchment, then dust with flour, tapping out excess. Whisk together milk, eggs, and vanilla.
In the bowl of a mixer fitted with the paddle attachment, beat together flour, sugar, baking powder, and salt on low speed until well combined. Continue beating while gradually adding butter until mixture is crumbly, about 3 minutes.
Slowly add half of milk mixture; increase speed to medium and beat until fluffy, about 2 minutes. Slowly add remaining half of milk mixture, scraping down bowl as needed. Beat until incorporated, about 1 minute more. Remove bowl from mixer; divide batter evenly among five bowls (about 1 heaping cup per bowl).
Tint four of the bowls with pink food color, starting with a small drop and gradually adding more to each bowl to create a gradient of shades.
Transfer batter from each bowl to one of the prepared pans, spreading to edges with a small offset spatula. Tap pans on counter to release any air bubbles. Bake until a tester inserted in centers comes out clean, about 15 minutes per cake. Transfer pans to wire racks; let cool completely. Turn cakes out of pans and remove parchment. In a small saucepan, heat jam and lemon juice and zest until warm.
Swiss Meringue Buttercream: Combine 1 1/2 cups sugar, 6 egg whites, and a pinch of salt in the heatproof bowl of a mixer set over (not in) a pot of simmering water. Whisk until sugar is dissolved and mixture is warm to the touch and feels completely smooth when rubbed between fingertips, 2 to 3 minutes.
Transfer bowl to mixer fitted with the whisk attachment; whisk on low speed until foamy. Increase speed to medium-high and whisk until stiff, glossy peaks form and mixture has cooled completely, about 10 minutes. Reduce speed to medium-low; add 4 sticks butter, 2 tablespoons at a time, whisking until fully incorporated after each addition. (Don't worry if buttercream appears curdled after all of butter has been added; it will become smooth again with beating.) Whisk in 1 teaspoon vanilla.
Switch to paddle attachment; beat on low speed until air bubbles are eliminated, 3 to 5 minutes. Transfer to a large bowl, cover with plastic, and let stand at room temperature. Repeat process twice more with remaining ingredients, using the same amounts each time, then combining all three batches of buttercream in the large bowl and covering with plastic.
Add pink food color, a drop at a time, until desired shade of light pink is reached.
To assemble cake, place darkest layer, bottom-side down, on an 8-inch cardboard cake round. Using a pastry brush, brush cake with a thin layer of jam mixture; top with 1 cup buttercream, spreading evenly to cover completely. Repeat process with three more layers, stacking darkest to lightest. Finish with lightest cake layer, bottom-side up. Evenly spread top and sides of cake with a 1/2-inch thickness of buttercream, smoothing sides with an offset spatula. Refrigerate until chilled, about 20 minutes.
Transfer remaining buttercream to a pastry bag fitted with a large petal tip (such as Wilton #127). Hold bag perpendicular to top of cake, with the wider part of the tip opening at the bottom. Pipe a petal in a circular pattern so that it adheres to top of cake. Simultaneously, rotate cake, and continue piping petals so that the end of each petal overlaps the beginning of the previous petal, until top of cake is covered completely. Next, pipe petals onto sides of cake: hold bag perpendicular to side of cake and pipe around cake, slightly overlapping each layer of frosting as you work from top to bottom of cake to cover completely.
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