Instructions

for the blackberry jam

 Process blackberries in food processor until smooth, about 1 minute; transfer to large saucepan. Stir sugar into blackberries and bring to boil over medium-high heat. Boil mixture, stirring often and adjusting heat as needed, until thickened and measures 1½ cups, 15 to 20 minutes. Transfer jam to bowl and let cool completely. (Jam can be refrigerated for up to 1 week; stir to loosen and bring to room temperature before using.)

for the cake

 Adjust oven rack to lower-middle position and heat oven to 325 degrees. Lightly grease two 8-inch round cake pans, line with parchment paper, grease parchment, and flour pans. Whisk 1¼ cups flour, ¾ cup sugar, 1½ teaspoons baking powder, and ¼ teaspoon salt together in large bowl. Whisk in 2 eggs and 3 yolks, 6 tablespoons butter, 2 tablespoons water, 1 teaspoon lemon zest and 2 tablespoons juice, and 2 teaspoons vanilla until smooth.
 Using stand mixer fitted with whisk attachment, whip 3 egg whites 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 ¼ cup sugar and whip until glossy, soft peaks form, 1 to 2 minutes. Whisk one-third of whites into batter to lighten. Using rubber spatula, gently fold remaining whites into batter in 2 batches until no white streaks remain.
 Divide batter evenly between prepared pans and smooth tops with rubber spatula. Bake until toothpick inserted in center comes out clean, 15 to 20 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. Repeat steps 2 through 5 with remaining cake ingredients to make two more cake layers.

for the mascarpone frosting

 Sprinkle gelatin over water in bowl and let sit until gelatin softens, about 5 minutes. Microwave mixture in 5-second increments until gelatin is dissolved and liquefied.
 Using clean, dry mixer bowl and whisk attachment, whip mascarpone, cream cheese, sugar, vanilla, and salt on medium speed until light and fluffy, about 30 seconds, scraping down bowl as needed; transfer to large bowl. Using clean, dry mixer bowl and whisk attachment, whip cream on medium-low speed until foamy, about 1 minute. Increase speed to high and whip until soft peaks just begin to form, about 1 minute, scraping down bowl as needed. Slowly pour in gelatin mixture, and continue to beat until stiff peaks form, about 1 minute. Using rubber spatula, stir ⅓ of whipped cream into mascarpone mixture to lighten; gently fold remaining whipped cream into mixture in 2 batches. Stir room temperature blackberry jam to loosen, then gently fold jam into mascarpone mixture until combined. Refrigerate frosting for at least 20 minutes or up to 24 hours before using.
 Line edges of cake platter with 4 strips of parchment to keep platter clean and place small dab of frosting in center of platter to anchor cake. Place 1 cake layer on platter. Spread 1 cup frosting evenly over top, right to edge of cake. Repeat with 2 more cake layers, pressing lightly to adhere and spreading 1 cup frosting evenly over each layer. Top with remaining cake layer and spread 1½ cups frosting evenly over top. Spread remaining frosting evenly over sides of cake to cover with thin coat of frosting. Run edge of offset spatula around cake sides to create sheer veil of frosting. (Cake sides should still be visible.) Refrigerate cake for 20 minutes. Garnish with blackberries and carefully remove parchment strips before serving. (Cake can be refrigerated for up to 24 hours; bring to room temperature before serving.)