How to Make It
Preheat oven to 350°F. Generously butter and flour a 14-cup Bundt or 10-inch tube pan. Sift together cake flour, baking powder, and salt.
Place 2 1/2 cups of the sugar in a large mixing bowl or the bowl of a stand mixer. Sprinkle with lemon zest, and, using your fingers, rub into sugar. Add butter and yogurt, and beat with an electric mixer on high speed until light and fluffy, 5 to 8 minutes. (It will look broken and curdled at first, but that’s okay.)
Add egg yolks and whole eggs, 1 at a time, beating well after each addition. Stir in vanilla. Gradually add flour mixture, beating on low just until incorporated. Pour batter into prepared pan. (Lift the pan a couple inches off the counter, and let it drop. Do this a few times to get rid of any air bubbles in the batter.)
Bake in preheated oven on center rack until a long wooden pick inserted in center comes out clean, 50 to 60 minutes, rotating pan after 40 minutes. Cool cake in pan on a wire rack 15 minutes.
While the cake bakes, bring water, lemon half, cardamom pods, and remaining 1/2 cup sugar to a simmer in a saucepan over medium-high. Simmer, stirring occasionally, until sugar dissolves, about 8 minutes. Remove from heat; cover pan until ready to use.
After cake cools 15 minutes, invert onto a rimmed baking sheet, and remove pan. (Cake will be upside down.) Using a thin knife or wooden skewer, make about 16 slits or holes all over bottom of cake. Remove lemon half and cardamom pods from lemon-cardamom liquid, and discard. Gradually pour lemon-cardamom liquid over bottom of cake, 1/4 cup at a time, allowing it to soak into cake after each addition. Cool cake completely before inverting onto a serving platter, about 2 hours.
Whip cream until light and airy. Serve cake with Pomegranate Syrup and whipped cream.