Add remaining 1/4 cup of milk and strawberry puree and blend on medium speed until butter is mixed in.
Add the egg white/milk/vanilla mixture a little at a time (divide it into about 3 additions), mixing well at medium speed for about 20 seconds after each addition. (Batter should be smooth, but don't over mix!)
In the bowl of an electric mixer, beat butter on medium speed for 8 minutes. Butter will become very pale and creamy.
Add powdered sugar, a little at a time, mixing after each addition.
Add vanilla extract and strawberry puree, mix on low until combined. Then turn mixer up to medium and beat for 6 minutes. Frosting will become very light & fluffy. (You can add a little more powdered sugar or a little more pureed strawberries to thicken/thin the frosting if necessary. Be very careful about adding in too much strawberry puree, a little at a time is best).
*To puree strawberries: place washed strawberries (with stems removed) in food processor and puree until smooth.
To assemble cake: Place one cake layer on a cake plate/stand. Add a generous dollop of frosting and smooth over layer. Place second cake layer on top of the frosting, and a generous dollop of frosting and smoother over cake layer. Place the third cake layer (placing the layer on upside down will give you a “flat top”) on top of the frosting, add generous dollop of frosting and smooth a thin layer on top and down sides of cake (crumb coat). Place cake in refrigerator to chill, just until the frosting is set about 20 minutes. Add remaining frosting to cake, smoothing top and down sides of cake.
Recipe adapted from Sweetapolita.