Whisk the flour, baking powder, baking soda, and salt in a medium bowl for 30 seconds, until evenly pale with no clumps.

Rub the lemon zest into the 1 1/2 cups of sugar in a large bowl for 1 minute. The sugar should look damp and smell strongly of lemon.

Add the softened butter and beat on medium-high speed for 3 to 4 minutes, until pale and fluffy like whipped frosting.

Beat in the eggs one at a time, 30 seconds each, until the batter looks smooth and glossy.

Stir the buttermilk and 1/4 cup lemon juice together. Add the flour mixture in three parts and the buttermilk mixture in two, starting and ending with flour, mixing on low speed until no dry flour shows.

Scrape the batter into the pan and smooth the top. Tap the pan twice on the counter until the surface looks level.

Bake for 35 to 40 minutes, until the top looks light golden, the edges pull from the pan, and a toothpick comes out with a few moist crumbs.

While the cake bakes, simmer 1/4 cup lemon juice and 1/4 cup sugar over medium heat for 2 to 3 minutes, until the liquid looks clear with no sugar crystals left.

Cool the cake in the pan for 10 minutes, then poke holes every inch with a skewer. Brush the warm syrup on slowly, until the surface looks glossy but not pooled.

When the pan bottom feels barely warm, about 35 minutes later, whisk the powdered sugar and 3 tablespoons lemon juice until the glaze falls from the whisk in a thick ribbon. Pour it over the cake, spread it to the edges, and wait 10 minutes until it turns matte. Cut into 15 squares.