Heat the oven to 350 °F and grease and flour a standard 4.5- or 5-inch by 8.5- or 9-inch loaf pan. Warm cold eggs in a bowl of hot water before mixing so they blend smoothly with the butter.

Place the sugar in the stand-mixer bowl and zest the lemons directly over it, then rub the zest through the sugar with your fingertips. The sugar should smell sharp and floral, with tiny yellow flecks distributed throughout.

Add the salt, vanilla, and room-temperature butter, then beat with the paddle attachment, beginning on low and increasing to medium speed. After at least 5 minutes, the mixture should look pale ivory, fluffy, and noticeably fuller in volume.

Reduce the mixer to low and add the eggs one at a time, scraping the bowl as needed between additions. Continue mixing for about 3 to 5 minutes until the batter looks cohesive and glossy rather than curdled.

Sift in the cake flour, then add the baking powder and sour cream; mix on low only until the batter is evenly combined. Stop once no dry streaks remain, since extra mixing can make the finished crumb rubbery.

Transfer the batter to the prepared pan and level the surface with a rubber spatula, then draw a shallow line down the center. The groove encourages the loaf to open along the middle as it rises.

Bake for about 50 to 60 minutes, depending on the loaf-pan size. The center should resist gently and spring back beneath a fingertip, and a cake tester should come out clean.

Near the end of baking, combine the lemon juice, sugar, and salt in a small saucepan and warm over medium heat just until the sugar dissolves. Keep the syrup below a boil so the lemon flavor stays fresh and bright.

While the cake is still hot in the pan, poke holes across the surface with a cake tester and drizzle the warm syrup over it gradually. Pause between pours so the crumb can draw in the syrup instead of letting it pool on top.

Let the loaf cool in the pan as it absorbs the syrup, then serve it warm or at room temperature. The cake should slice cleanly once cooled, with a moist crumb and a lightly glossy top.