For the Small Batch Strawberry Cake

• First, make the sheet cake: Position a rack in the center of the oven and preheat the oven to 350°F. Lightly spray a 9 x 5-inch loaf pan with cooking spray and line with parchment paper, leaving a 2-inch overhang on the pan's two long sides. Spray the parchment, too.
• Pour the milk into a small bowl or liquid measuring cup. Scoop out 1 tablespoon into another small bowl. Add the egg white and vanilla to the bowl with the small amount of milk and whisk to combine.
• In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, baking powder, and salt. Beat on low until just combined, about 15 seconds. Add the butter all at once and beat on low until the mixture has the texture of coarse meal, with pea-sized pieces of butter throughout, about 3 minutes. Add the first bowl of plain milk all at once and increase the mixer to medium. Beat until the batter is light and fluffy, about 2 minutes, using a rubber spatula to scrape down the bottom and sides of the bowl as necessary. Reduce the mixer to low and add the milk and egg white mixture in two or three additions, scraping down the bottom and sides of the bowl after each addition, then beat until just combined.
• Pour the batter into the prepared pan. Bake for 25 to 30 minutes, or until the top of the cake bounces back when gently pressed. A skewer inserted into the cake should come out with a few crumbs attached. Cool completely in the pan on a wire rack before frosting.
• Next, make the strawberry buttercream frosting: In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium-low until soft and creamy, 2 to 3 minutes. Scrape down the bottom and sides of the bowl with a rubber spatula. With the mixer on low, gradually add the confectioners' sugar and beat until combined. Scrape down the bottom and sides of the bowl. With the mixer on low, add the strawberry puree, beating just to incorporate the liquid—if it still seems dry, add the milk, 1 teaspoon at a time, until creamy. Scrape down the bottom and sides of the bowl once more, then beat on medium-high until the frosting is creamy and smooth, 2 to 3 minutes.
• Use an offset spatula to cover the top of the cake with the frosting completely. Garnish with fresh strawberries. The cake will keep in an airtight container for 1 day. After that, transfer to the refrigerator and refrigerate for up to 2 more days.