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.
Nothing says summer quite like this easy but remarkably grand strawberry dessert. Even better, it can wait patiently in the refrigerator for up to a day.