To prepare the cake, preheat oven to 350°F. Line two 8- or 9-inch round cake pans with parchment paper. Butter and flour the parchment paper to keep the cakes from sticking.
In bowl of electric mixer, beat butter and sugar on high speed until pale and creamy, about 3-4 minutes, scraping down bowl as necessary.
In medium bowl, combine cake flour, baking powder and salt. In another bowl, whisk together milk, eggs and vanilla extract. Add flour mixture to butter mixture in 3 additions, alternating with milk mixture and ending with flour mixture. Scrape down bowl between each addition.
Divide batter between two prepared cake pans and bake until tops are barely golden brown and spring back when lightly touched, about 30-35 minutes. Let cool in pan 10 minutes, then transfer to wire rack to cool to room temperature. Cover cake layers with plastic wrap and refrigerate for at least 1 hour and up to 2 days.
When ready to assemble cake and serve, prepare the strawberries. Toss sliced strawberries, sugar and vanilla extract in bowl and let sit for 20 minutes.
To prepare whipped cream, use electric mixer to beat whipping cream, sugar and vanilla extract to medium peaks.
Remove chilled cake layers from refrigerator and use small knife to score each layer in half horizontally. Then use a large serrated knife to slowly cut all the way through each layer, using scored line as a guide.
Place one cake layer on cake plate or stand. Cover with 1 cup whipped cream and one quarter of strawberry mixture. Repeat with remaining cake layers, whipped cream and strawberries.
Serve cake immediately.