Preheat oven to 325 degrees, with rack positioned in lower third of oven. Sift flour and 3/4 cup sugar into a medium bowl. Repeat sifting four times.
Beat egg whites and the warm water with a mixer on low speed until foamy. Add salt, cream of tartar, and vanilla. Increase speed to medium; beat until soft peaks form, about 3 minutes. With mixer running, add remaining 3/4 cup sugar, 1 tablespoon at a time, beating well after each. Increase speed to high; beat until stiff, glossy peaks form (egg-white mixture should not be dry), about 2 minutes. Transfer to a large bowl. Sift flour mixture over egg-white mixture in 6 additions, folding gently after each.
Pour batter into a pastry bag. Cut off tip, and pipe batter into 1-cup mini angel food cake pans, filling each 3/4 full. Gently run the tip of a knife or skewer through batter to remove air bubbles that may have formed when piping. Bake until cakes are golden brown and spring back when touched, 25 to 30 minutes. Place cake pans upside down on a wire rack, and let cakes cool for 1 hour. Carefully run a paring knife around sides of cakes to loosen, then flip over to remove from tins. Cakes will keep, covered, for 1 day.
Meanwhile, make glaze: Whisk together confectioners' sugar and lemon juice in a small bowl until smooth. If not using immediately, cover with plastic wrap, pressing it directly on the surface, and let stand at room temperature; whisk until glaze is smooth before using.
To assemble: Spoon some glaze over each cake. Top with spun sugar, and serve immediately.
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