Tender, lemony butter cake lies beneath a crisp, crunchy, crackling shell for a delicious contrast in every bite of this cake.
Preheat oven to 350 degrees. Brush an 8-by-2-inch square cake pan with butter. Line with parchment, leaving a 2-inch overhang on all sides; brush parchment with butter and dust with flour, tapping out excess.
Whisk together flour, baking powder, and salt in a bowl. Stir together milk and vanilla in another bowl. Beat butter on medium-high speed until pale and fluffy, about 3 minutes. Add 1 1/3 cups granulated sugar in a steady stream and beat, scraping down sides of bowl as needed, until pale and fluffy, about 3 minutes. Beat in finely grated lemon zest and juice.
Reduce speed to low. Add flour mixture in 3 batches, alternating with milk mixture; beat until just combined.
In a large bowl, whisk egg whites until soft peaks form. Fold half into batter, then fold in remainder. Transfer to prepared pan, smoothing top. Sprinkle with remaining 1/4 cup granulated sugar, julienned lemon zest, and sanding sugar.
Bake until a tester inserted in center comes out clean, about 40 minutes. Let cool in pan on a wire rack 30 minutes. Using parchment, carefully transfer to rack; let cool completely. Slice with a serrated knife, dollop with whipped cream, and serve.