Special equipment: a 10-inch tube pan (not the kind with a removable bottom), a long skewer
1. For the cake: Preheat the oven to 350 degrees F. Separate the eggs, putting the whites in one large bowl and the yolks in another.
2. Add the cream of tartar to the whites, and beat with an electric mixer on medium-high until soft peaks form, about 3 minutes. Reduce the speed to low, and gradually add 1/3 cup of the sugar. When all the sugar is in, turn the speed back up to medium-high and continue to beat until stiff, glossy peaks form, about 4 minutes.
3. Add the remaining sugar and 1 tablespoon of the vanilla to the bowl with the yolks, and beat with the electric mixer until the mixture is pale yellow and very thick and a bit of the batter dropped on top of the rest sits there for a few seconds before sinking in, about 5 minutes. Sift the flour, baking powder and salt over the mixture, and fold until blended. Fold in about 1/3 of the whites until the batter has lightened in texture. Fold in the remaining whites in two additions until smooth and no streaks of white remain. Pour the batter into an ungreased 10-inch tube pan.
4. Bake until the cake is browned and the top springs back to the touch, about 35 minutes. Immediately turn the pan over onto a cooling rack, and let the cake cool completely in the pan. Run a thin spatula around the outside edge and the inside ring; gently working around the cake, pull with a spatula until it releases from the pan. Scrape out the crumbs from the bottom and sides of the pan (it doesn't have to be totally clean, but you don't want the crumbs to absorb the soaking milks), then put the cake back in the pan.
5. Using a long skewer, poke holes all over the top of the cake (around 60 is good), going all the way to the bottom of the pan. Whisk together the sweetened condensed milk, evaporated milk, cream and remaining 1 teaspoon vanilla in a medium bowl. Slowly pour the mixture over the cake, letting it run down the sides of the cake, making sure the cake is well coated. (It may seem like a lot of liquid, but be sure to use it all so the cake is moist throughout.) Cover and refrigerate for at least 6 hours or overnight.
6. One hour before serving, remove the cake from the refrigerator and let stand at room temperature to warm slightly.
7. For the frosting: Meanwhile, bring a few inches of water to a simmer in a medium saucepan. Set a large stainless-steel bowl over it (do not let the bowl touch the water); add the sugar, egg whites, 1/4 cup cold water, vanilla, cream of tartar and salt, and whisk until the ingredients are combined. Then, with the bowl still over the simmering water, beat with an electric mixer on medium-high speed until thick, glossy, stiff peaks form, 5 to 7 minutes. Remove the bowl from the heat, and let stand a few minutes.
8. To assemble: Run a thin spatula around the outer edge and the inside rim of the pan, gently pulling to release the cake. Turn it out onto a rimmed serving plate. Generously frost the top, the sides and the inside ring, using a small offset spatula or spoon. Microwave the dulce de leche in a small microwave-safe dish until it's loose enough to drizzle, about 8 seconds. Drizzle the cake with the dulce de leche, letting it gently drip down the sides.
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