Line the bottom of an 8-inch round cake pan (with 2- to 3-in.-tall sides) with cooking parchment cut to fit. If pan has a removable rim, wrap outside with foil to prevent water from seeping in.
In a large bowl, with a mixer on high speed, beat whole eggs, egg white, and sugar until thick and pale yellow, 4 to 5 minutes. Stir in vanilla. In a small bowl, mix flour and baking powder until blended. With a whisk, fold flour mixture into egg mixture until evenly blended. Scrape batter into pan. Tap pan lightly on work surface to remove air pockets.
Set a round rack (cake rack, basket steamer, removable rim from a cake pan, or empty 2-in.-tall cans with both ends cut out) in a 12- to 14-inch steamer, 14-inch wok, or 12-inch frying pan (with at least 2-in.-tall sides; steamer should be wide enough to hold cake pan without sides touching). Pour water into steamer to about 1/2 inch below top of rack. Cover steamer and bring water to a boil over high heat.
Place cake on rack and cover steamer (if cover doesn't fit over cake, tent steamer with a large piece of foil, sealing it around the edges so steam can't escape); reduce heat to medium and steam until a toothpick inserted in center of cake comes out clean, 20 to 25 minutes. If necessary, add more boiling water to maintain level. Remove cover quickly and carefully so condensed steam doesn't drip onto cake. Remove cake from steamer.
Run a knife between cake and pan rim to loosen sides. Invert onto a rack, remove pan and parchment, and invert again onto another rack. Let cake cool at least 10 minutes. With a serrated knife, cut cake into diamonds or wedges. Serve warm or cool.