Preheat the oven to 350°F (175°C). Grease your cake pans with butter or nonstick spray, then line the bottoms with parchment paper. In a medium bowl, whisk together the flour, baking powder, and salt until evenly combined. Set aside.
In a large bowl (or the bowl of your stand mixer), beat the softened butter and sugar together on medium speed for a full 3 minutes. Stop and scrape the sides once. The mixture should look pale and fluffy — this is where the cake gets its lightness. I’ve found that setting a timer prevents me from rushing.
Crack the eggs in one at a time, beating well after each addition. The batter might look a little curdled at first; that's normal. Beat in the vanilla. The mixture should be thick and glossy.
With the mixer on low, add about one-third of the flour mixture. Then half the milk. Another third of flour. The rest of the milk. End with the remaining flour. Mix only until the streaks disappear — overmixing develops gluten and makes the cake tough. Scrape the bowl once during this process.
Divide the batter evenly between the two pans. Bake for 25–30 minutes, rotating halfway through. The tops will be golden, the edges will pull away from the sides, and a toothpick inserted in the center should come out clean or with just a few moist crumbs. Cool the cakes in the pans for 10 minutes, then turn them out onto a wire rack and let them cool completely. This is important: if the cake is even slightly warm, the frosting will melt.
Just before you're ready to assemble, pour the cold heavy cream into a chilled bowl. Whip on medium-high until stiff peaks form. Gently fold in the peach purée with a rubber spatula — fold just until incorporated. Overfolding will deflate the cream. If it seems too soft, pop it in the fridge for 10 minutes to firm up.
If you want a dramatic four-layer cake, use a serrated knife to slice each cake layer in half horizontally. Place the first layer on your serving plate. Spread about ¼ of the peach cream evenly over the top. Repeat with remaining layers. For the top, spread the last of the cream, letting it cascade slightly over the sides for a rustic look. Arrange fresh peach slices on top.