Heat the oven to 350°F (175°C) and line a 12-cup muffin pan with paper liners. Sift the cake flour, baking powder, salt, and 1/2 cup of the sugar into a medium bowl.

In a separate bowl, whisk together the three egg yolks, oil, fresh lemon juice, lemon zest, and vanilla extract. Stir this mixture into the dry ingredients just until smooth, taking care not to overmix.

Place the four room-temperature egg whites and cream of tartar in a clean, grease-free bowl. Beat until foamy, then gradually add the remaining 1/4 cup sugar and continue beating until medium-stiff, glossy peaks form.

Fold about one-third of the whipped egg whites into the lemon batter to lighten it. Gently fold in the remaining whites in two additions, stopping as soon as no large white streaks remain.

Divide the batter among the paper liners, filling each about two-thirds full. Bake for 18–22 minutes, until the tops spring back lightly when pressed and a toothpick inserted in the center comes out clean.

Cool the cupcakes in the pan for 5 minutes, then transfer them to a wire rack. Cool completely for about 30 minutes before adding frosting.

For the optional frosting, beat the softened butter until smooth. Gradually add the sifted confectioners’ sugar, lemon juice, zest, and salt, then beat until light and fluffy, adding milk or cream a little at a time if needed.

Pipe or spread the frosting onto the cooled cupcakes. Finish with a little finely grated lemon zest or a tiny lemon slice just before serving.