Prep the oven, pan, and water bath. Position a rack in the center of the oven and preheat the oven to 350°F. Lightly butter an 8 x 8-inch ceramic or glass square cake pan. Bring a large kettle of water (enough to fill the roasting pan with 1 to 2 inches of water) to a boil. Make the lemon sugar. In a large glass bowl, combine the sugar and lemon zest. Use your fingers to rub the zest into the sugar—this will infuse the sugar with oils from the zest.
Mix in the rest of the ingredients (except the egg whites). Add the lemon juice, buttermilk, flour, butter, salt, and egg yolks and mix to combine. It’s okay if the mixture looks lumpy—that’s totally normal, I promise!
Whisk the egg whites. In the bowl of a stand mixer fitted with the whisk attachment, whisk the egg whites. Whisk on low speed until foamy, 1 minute, then turn the mixer up to medium-high. Whisk until stiff peaks form, 3 to 4 minutes.
Fold the egg whites into the rest of the ingredients. Use a wooden spoon or ladle to carefully scoop the egg whites into the medium bowl with the rest of the ingredients—be careful not to deflate them! Use a rubber spatula to fold in the ingredients until just combined.
Assemble the cake for baking. Pour the batter into the prepared cake pan and use an offset spatula to smooth the top. Set the cake pan inside the roasting pan. Pour the boiled water into the pan until it reaches halfway up the sides of the cake pan.
Bake the cake. Bake for 40 to 45 minutes, or until the top of the cake bounces back when gently pressed. The cake should wobble ever so slightly when pulled out of the oven.
Serve and store. Serve warm, or at room temperature. The lemon pudding cake will keep, covered loosely in plastic wrap, in the refrigerator for up to 3 days. To reserve, rewarm in the microwave or eat chilled.