Simmer blackberries, sugar, and 1/4 cup water in a medium saucepan over medium-high heat until the blackberries are soft and very saucy, about 5 minutes. Remove from the heat, stir in the lemon juice and zest. Pour into a pie plate and let cool to room temperature.
Spray a 3 1/2-quart metal or glass bowl with cooking spray and line with plastic wrap, leaving a 1-inch overhang over the edge of the bowl.
Spread 2 cups of vanilla ice cream over the bottom of the bowl. Use a large piece of plastic wrap to press the ice cream down into the bowl. Dip both sides of half the angel food cake in to the blackberry syrup (like you do with French toast) and arrange, on top of the ice cream, tearing pieces of cake as needed to cover in a flat layer. Top with any remaining sauce.
Spread the raspberry sorbet over the angel food cake. Use a large piece of plastic wrap to press down the sorbet. Flatten with a spatula and top with the remaining angel food cake slices, tearing pieces of cake as needed to cover in a flat layer. Fold the overhanging plastic over top and gently press down. Freeze until hard, at least 4 hours up to overnight.
Make the meringue: Whip the egg whites and cream of tartar in a large bowl with an electric mixer at medium-high speed until foamy, about 2 minutes. Increase the speed to high, gradually add the sugar and whip until the whites hold stiff peaks and the sugar has dissolved, 5 minutes more. Invert the ice cream cake onto a parchment-lined baking sheet, remove the plastic wrap. (If the cake is difficult to remove, let the bowl stand overturned until it slips out easily, about 5 minutes.) Spread the meringue evenly over the sides, but slightly thicker on top. Use the back of a spoon to make large swirly peaks. Freeze 3 hours or up to 2 days.
Preheat the oven to 500 degrees. Bake the dessert until the meringue is browned, 3 to 5 minutes. Alternately, use a small torch to brown the meringue all over. Use two large spatulas to transfer the baked Alaska to a large serving plate, cake stand or cutting board. Slice into wedges.
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