In a deep bowl, beat egg whites and cream of tartar with a mixer on high speed until frothy. Add 1 cup sugar, about 1 tablespoon every 30 seconds, beating whites until they hold stiff peaks. Beat in vanilla and anise.
Lightly butter and flour-dust an 8-inch cheesecake pan with removable rim. Spread meringue level in pan. Bake in a 275° oven until pale golden, 1 1/4 to 1 1/2 hours.
Run a thin-bladed knife between meringue cake and pan rim to release. Cool cake on a rack, about 20 minutes.
Thinly slice peaches; mix 2 cups slices with 1/2 cup raspberries, 1 tablespoon sugar, lemon peel, and 1 tablespoon lemon juice.
In a blender or food processor, whirl smooth remaining peaches, raspberries, 2 tablespoons sugar, and 2 tablespoons lemon juice. Rub puree through a fine strainer into a bowl; discard residue.
Remove pan rim and set meringue cake on a plate. Spoon sliced peaches and whole berries onto cake and drizzle with about 2 tablespoons fruit sauce. Cut cake into wedges and add remaining sauce to taste.