For the cake: Prepare an ice-water bath. Combine lemon juice, 3 cups blueberries, 1/4 cup granulated sugar, and 1/4 teaspoon salt in a medium saucepan over medium heat. Cook, stirring gently, until sugar dissolves and juices thicken, about 8 minutes. Transfer to a small bowl, and set in ice-water bath, stirring occasionally, until cooled. Reserve 1/4 cup of cooled blueberry juice from the berries in a small bowl, cover, and refrigerate.
Preheat oven to 350 degrees. Butter an 18-by-13-inch rimmed baking sheet, line with parchment, and butter parchment. Whisk cornmeal, baking powder, 1 teaspoon salt, and 1/2 cup flour in a medium bowl. Cream butter and remaining 1 1/4 cups granulated sugar with an electric mixer until light and fluffy, 2 to 3 minutes. Add eggs, 1 at a time, mixing well to incorporate after each addition. Add vanilla, and scrape down sides of bowl. Lower the speed to the lowest setting, and beat in 1/2 the flour mixture, then the yogurt, and then the remaining flour mixture, mixing well after each addition.
Gently fold remaining 1/2 cup flour and 2 cups blueberries into the berry mixture, until well combined. Fold the berry-flour mixture into the cornmeal batter with a spatula, until uniformly purple. Spread evenly in baking sheet, and bake, rotating halfway, until the top is golden brown and the sides pull away from the pan, 28 to 30 minutes. Transfer baking sheet to wire rack, and let cool completely. Cut 24 circles using a 2 1/2-inch round cutter; discard scraps.
For the cheesecake: Line an 18-by-13-inch rimmed baking sheet with parchment leaving a 1-inch overhang. Beat the cream cheese with a mixer on medium-high until completely smooth, 4 to 5 minutes. Beat in confectioners' sugar, honey, yogurt, and salt, and beat until smooth, about 3 minutes. Gently whisk in 1/2 of the whipped cream, then fold in remaining whipped cream until well incorporated. Scrape batter onto baking sheet, and smooth. Freeze, uncovered, until firm, at least 1 hour (or up to 1 day, covered). With the same round cutter, cut out 24 circles, dipping cutter into hot water and wiping dry between cuts.
For the compote: Toss raspberries and blueberries with reserved 1/4 cup blueberry juice.
Place 1 cake round, browned side down, on a plate. (Trim cakes to make level, if necessary.) Top cake round with a frozen cheesecake round; repeat to make 4 cake-cheesecake layers. Refrigerate composed cakes while quickly finishing assembly of remaining cakes. Top each cake stack with berry compote. Serve immediately.