For the Cake Layers

1. Adjust oven rack to middle position and heat oven to 350 degrees. Grease two 9‑inch or three 8‑inch round cake pans, line with parchment paper, grease parchment, and flour pans. Whisk flour, baking powder, baking soda, salt, and 1 1/2 cups sugar together in bowl. Whisk buttermilk, melted butter, egg yolks, oil, and vanilla together in second bowl.
2. Using stand mixer fitted with whisk attachment, whip egg whites and cream of tartar on medium-low speed until foamy, about 1 minute. Increase speed to medium-high and whip whites to soft billowy mounds, about 1 minute. Gradually add remaining 1/4 cup sugar and whip until glossy, stiff peaks form, 2 to 3 minutes; transfer to third bowl.
3. Add flour mixture to now-empty mixer bowl and mix on low speed, gradually adding buttermilk mixture and mixing until almost incorporated (a few streaks of dry flour will remain), about 15 seconds. Scrape down bowl, then mix on medium-low speed until smooth and fully incorporated, 10 to 15 seconds.
4. Using rubber spatula, stir one-third of whites into batter. Gently fold in remaining whites until no white streaks remain. Divide batter evenly between prepared pans and smooth tops with rubber spatula. Gently tap pans on counter to settle batter. Bake until toothpick inserted in center comes out clean, 20 to 22 minutes, switching and rotating pans halfway through baking. Let cakes cool in pans on wire rack for 10 minutes. Remove cakes from pans, discarding parchment, and let cool completely on rack, about 2 hours. (Cake layers can be stored at room temperature for up to 24 hours or frozen for up to 1 month; defrost cakes at room temperature.)

For the Jam Filling

1. Process sugar, pectin, and salt in food processor until combined, about 3 seconds. Add blueberries and pulse until chopped coarse, 6 to 8 pulses. Transfer blueberry mixture to medium saucepan and bring to simmer over medium heat, stirring occasionally, until mixture is bubbling and just starting to thicken, 6 to 8 minutes. Off heat, stir in lemon juice. Transfer 11/3 cups jam to small bowl, cover, and refrigerate until firm, about 3 hours. Strain remaining jam through fine-mesh strainer set over bowl, cover, and set aside at room temperature. (You should have at least 1/4 cup.)

For the Frosting

1. Using stand mixer fitted with paddle, beat butter and sugar on medium-high speed until light and fluffy, about 3 minutes. Add cream cheese, 1 piece at a time, and beat until no lumps remain. Add vanilla and 2 tablespoons strained jam and mix until incorporated. Transfer 1/3 cup frosting to each of 2 small bowls. Add 1 teaspoon strained jam to first bowl and 1 tablespoon strained jam to second bowl, stirring well to combine. (You should have 3 shades of frosting.)
2. Place 1 cake layer on cake turntable. Spread 2/3 cup chilled jam evenly over top. Repeat with 1 more cake layer, pressing lightly to adhere, and remaining chilled jam. Top with remaining cake layer, pressing lightly to adhere. Spread small amount of lightest-colored frosting in even layer over top and sides of cake. Using small offset spatula, spread darkest-colored frosting over bottom third of sides of cake; medium-colored frosting over middle third; and remaining lightest-­colored frosting over top third. While spinning cake turntable, run spatula from bottom to top of side of cake to blend frosting colors. While spinning cake turntable, run spatula over top of cake, working from outside in to create spiral. Serve.