Preheat oven to 350°F. Grease the bottoms of three 8-inch round cake pans and line with parchment.
In the bowl of stand mixer with paddle attachment, combine the flour, sugar, baking powder, and salt on low speed. (You can also use a hand-held mixer)
Add in the butter one piece at a time. Beat on low speed until all the butter has been incorporated, about 3 minutes. The mixture should have a fine crumbly, cornmeal-like texture.
In a medium measuring glass with a spout, combine half the milk, the cherry juice, lemon juice, and vanilla.
In a separate measuring cup, gently whisk the egg whites and remaining milk.
With the stand mixer on low speed, gradually add the cherry juice mixture, followed by pink food color and beat for five minutes. Scrape the sides and bottom of the bowl with a rubber spatula.
Put the mixer back on low speed and add the egg white mixture in three parts, scraping down the sides of the bowl after each addition. Beat for 1 minute.
Toss the cherry bits in a little flour and gently fold them into the batter by hand.
Divide the batter evenly among the prepared pans.
Bake for 20-25 minutes or until a toothpick inserted into the center is removed with few crumbs.
Let layers cool in their pans on wire racks for 10 minutes. Using a knife, loosen the sides of the cakes and carefully turn them out onto wire racks. Peel off parchment layers and cool completely.
The cake layers will keep wrapped tightly in plastic wrap at room temperature for up to 2 days.