Toss the 4 cups mixed frozen berries with 3 tbsp sugar and 2 tbsp lemon juice. Cover and refrigerate up to 4 hours. They'll release a ton of juice on their own, no cooking needed.
Pull the 4 egg whites out 30 minutes ahead. Room temperature whites whip to noticeably more volume than cold ones.
Heat the oven to 335°. Line a 9x13 pan with parchment, bottom and sides.
Sift the 2 cups all-purpose flour, ⅓ cup cornstarch, 2-¼ tsp baking powder, ¼ tsp baking soda and ½ tsp salt together.
Blend the 1 cup cottage cheese, 4 egg yolks, ⅓ cup milk, ⅓ cup powdered sugar, 1-½ tsp vanilla extract and ¼ tsp almond extract until completely smooth. No curds, no specks. Whisk in the 2 tbsp butter and 2 tbsp avocado oil.
Fold the dry into the wet just until combined. Don't beat it, leave some flour streaks.
Whip the egg whites with ½ tsp cream of tartar to soft peaks, then stream in ½ cup sugar and whip to firm, glossy peaks. Stop before stiff.
Stir a third of the meringue into the base to loosen it. Fold in the rest in two additions, gently, until no white streaks remain.
Bake 22 to 26 minutes, until the center springs back. Cool completely in the pan, about an hour on a rack.
Set a sieve over a bowl and drain. You want about ½ cup of juice. Stir ¼ tsp almond extract into the juice and set it aside for the soak.
Slice off the domed top of the cake. Brush the reserved juice over the top and let it absorb 15 minutes. But only do this if you're ready to serve the cake in the next hour.
Stir the 2 tbsp chia seeds into the drained fruit and let sit 20 minutes to thicken.
Spread the chia berries over the cake in an even layer.
Beat the 8 oz whipped cream cheese, ½ cup cottage cheese and 1-¼ cups powdered sugar until completely smooth. In a separate bowl, whip the 1 cup heavy cream with 1-½ tsp almond extract, 1 tsp vanilla extract and a pinch of salt to firm peaks. Fold the whipped cream into the cream cheese mixture in two additions.
Spread over the berries.
Top with 1-½ cups fresh berries and ⅓ cup slivered almonds just before serving.