Preheat oven to 350 degrees. Butter bottom of an 8-inch round cake pan, and line with parchment paper. Butter and flour paper and sides. In a medium bowl, whisk together flour, baking powder, and salt; set aside.

Using an electric mixer on high speed, beat butter and 1/2 cup sugar in a large bowl until light and fluffy. Add eggs and yolks, one at a time, beating well after each addition. Beat in vanilla. With mixer on low, alternately add flour mixture in 3 parts and milk in 2, beginning and ending with flour mixture; mix just until combined. Spread batter in prepared pan.

Bake until a toothpick inserted in center of cake comes out clean, 30 to 35 minutes. Cool in pan 10 minutes; invert onto a wire rack to cool completely. Using a serrated knife, split cake in half horizontally; place bottom half, cut side up, on a serving plate.

Make topping: In a large bowl, combine strawberries and 1/4 cup sugar; set aside. Place 2 tablespoons cold water in a small saucepan, and sprinkle with gelatin; let soften 5 minutes. Place saucepan over very low heat, and stir until gelatin is dissolved. Remove from heat; let cool.

Using an electric mixer, beat cream and remaining 1/4 cup sugar in a large bowl until very soft peaks form. Continue to beat, and gradually add gelatin mixture; beat until soft peaks form.

Arrange half of strawberries over bottom cake layer; top with half of whipped cream, leaving a 1-inch border. Cover with top half of cake, cut side down. Top cake with remaining whipped cream, leaving a 1-inch border. Refrigerate cake and remaining strawberries separately, at least 1 hour (or up to 1 day). Just before serving, spoon strawberries over cake.