1. Lay chiles in a single layer on a 12- by 15-inch baking sheet. Bake in a 400° oven just until pliable, about 2 minutes. Wearing rubber gloves, break off stems, shake out seeds, and break chiles into small pieces, dropping into a small bowl; discard stems and seeds. Cover chiles with warm water and let soak until soft, 5 to 7 minutes. Drain chiles and put in a blender with 1/3 cup water; whirl until smooth, adding 1 more tablespoon water as needed to make a thick paste. Push purée through a fine strainer; discard residue. You need 1/3 cup chile purée. If using ancho chiles, stir cayenne into the chile purée.
2. Line bottom of a 9-inch cake pan (sides at least 1 1/2 in. tall) with baking parchment.
3. In a large bowl nested over a pan of simmering water (water shouldn't touch bottom of bowl), combine chocolate and butter. Stir occasionally just until chocolate is melted and mixture is smooth, about 8 minutes. Remove from over water and whisk in 1/3 cup chile purée, the egg yolks, vanilla, and flour until mixture is blended.
4. Pour brown sugar into a small bowl and stir or whisk to break up lumps and loosen. In a large bowl, with an electric mixer on high speed, beat egg whites and cream of tartar until very frothy and foamy. Gradually add brown sugar to whites, beating until stiff, moist peaks form. With a whisk, fold a third of the beaten whites into the chocolate mixture until well incorporated. Then fold in remaining whites just until blended. Scrape batter into prepared pan.
5. Bake cake in a 425° regular or 400° convection oven until it appears set and center barely jiggles when pan is gently shaken, about 15 minutes. Let cool in pan on a rack for about 15 minutes. Run a knife between cake and pan rim, then invert onto a serving platter. Lift off pan and peel off parchment. Let cake cool about 30 minutes, then chill until firm and cold, at least 4 hours; cover cake once completely chilled.
6. For best texture, let cake come to room temperature before serving, 45 minutes to 1 hour. Sift powdered sugar lightly over cake (for a pattern, lay a stencil on cake before sifting the sugar, then carefully lift it off).
7. In a bowl, beat whipping cream until soft peaks form. Stir in vanilla. Cut cake into wedges and serve each with a dollop of whipped cream.