Wash, stem, and quarter about 2 cups of fresh strawberries, then blend until smooth. Strain through a fine-mesh strainer if desired, and measure out 1 cup for the cake and ½ cup for the frosting.
Preheat oven to 350°F (175°C). Grease two 9-inch round cake pans, line the bottoms with parchment paper, then lightly dust with flour and tap out any excess.
In a medium bowl, whisk together the flour, baking powder, and salt until evenly combined; set aside.
Beat softened butter on medium speed for 2 minutes until smooth, then add sugar and beat on medium-high for 3–4 minutes until light, fluffy, and pale, scraping the bowl as needed.
Add eggs one at a time, beating well after each addition, then mix in the vanilla extract until fully incorporated.
On low speed, alternate adding the flour mixture and the strawberry puree with milk in three additions of flour and two of liquid, beginning and ending with flour; mix just until no streaks remain and do not overmix.
Divide batter evenly between the prepared pans, smooth the tops, and tap on the counter to release air bubbles. Bake for 28–32 minutes, until a toothpick inserted in the center comes out with just a few moist crumbs.
Cool cakes in pans on wire racks for 10 minutes, then run a knife around the edges, invert onto racks, peel off parchment, and cool completely before frosting.
Beat butter for 2–3 minutes until creamy, then gradually add powdered sugar, followed by the strawberry puree and vanilla; beat on medium-high for 3–4 minutes, adding heavy cream one tablespoon at a time until the desired consistency is reached.
Place the first layer on a cake stand and spread about 1 cup of frosting on top, then add the second layer, apply a thin crumb coat, refrigerate 15 minutes, and finish with the remaining frosting. Garnish with fresh strawberry slices.