Preheat oven to 350°F. Coat 3 (9-inch) round cake pans with baking spray. Beat butter with a stand mixer fitted with paddle attachment on medium speed until creamy, 2 to 3 minutes, stopping to scrape down sides as needed. With mixer running on medium speed, gradually add 2 cups of the granulated sugar, beating until light and fluffy, 3 to 5 minutes, stopping to scrape down sides of bowl as needed. Add egg yolks, 1 at a time, beating until just combined after each addition.

Whisk together flour, baking powder, and salt in a medium bowl. With mixer running on low speed and beating until just combined after each addition, add flour mixture to butter mixture, alternately with milk, beginning and ending with flour mixture. Beat in 2 teaspoons of the vanilla.

Beat egg whites with an electric mixer on high speed until stiff peaks form, about 3 minutes. Working in batches, fold egg whites into batter until just combined. Spoon batter into prepared pans; spread in an even layer.  

Bake in preheated oven until a wooden pick inserted in center comes out clean, 18 to 20 minutes. Cool in pans on wire racks 10 minutes; remove from pans, and cool completely on wire racks, about 1 hour.

Gently stir together peaches and remaining 1/2 cup granulated sugar in a large bowl. Let stand, gently stirring occasionally, 30 minutes. Reserve 1/2 cup drained peaches for garnish.

Whisk together mascarpone cheese, powdered sugar, and remaining 2 teaspoons vanilla in a large bowl. Beat heavy cream with a stand mixer fitted with whisk attachment on medium-high speed until stiff peaks form, about 3 minutes. Gently fold into mascarpone mixture.

Trim 1/8-inch off top of each cake layer. (This helps layers to soak up peach juices.) Place 1 cake layer, cut side up, on a serving plate, and top with one-third of remaining peach slices and juice in even layer. Carefully spread one-third of whipped cream mixture over peaches. Repeat layers twice. Top with reserved 1/2 cup peach slices. Serve immediately or cover and chill up to 24 hours.