Cream together the butter, powdered sugar, vanilla extract, and salt in the bowl of a stand mixer with the paddle attachment until light and fluffy. About 3-4 min on medium speed.
Scrape the sides and bottom of the bowl. Then add in the whole egg and beat again until smooth and fluffy. You may need to scrape the sides of the bowl again for everything to incorporate fully.
Adding in the Dry Ingredients
Now add the flour to the mixing bowl and mix on medium low speed until smooth and everything is fully combined. The dough should not be crumbly anymore.
Once fully mixed, remove 350 grams of the cookie dough, shape it into a square and wrap it in plastic wrap.
In a food processor or a high speed blender, add in the strawberries and blend to a fine powder.
Add the strawberry powder into the mixing bowl with the remaining dough. Then add in 2 tablespoons of heavy cream and mix again until smooth. If the dough seems dry, add in another tablespoon of cream.
Once fully mixed, shape the strawberry dough into a square and wrap in plastic.
Place both pieces of dough in the fridge to chill for 45 min.
Shaping & Baking
Preheat the oven to 350 F (325 F if convection)
On a lightly floured surface, roll out both pieces of dough into rectangles that are 30 cm x 20 cm.
In a small bowl, gently whisk the egg white with a fork until it is frothy.
Brush the frothy egg white onto the vanilla dough. Then in one smooth motion lay the strawberry dough on top of the vanilla dough.
At one of the short ends, carefully trim away 2 cm of the strawberry dough. This will be the end that you start the roll from.
Then trim the opposite short end of both doughs so they are even and straight.
Roll the two layers into a spiral starting at the short end with the uneven layers. The vanilla dough will have a tendency to crack - so roll slowly and squeeze together any cracks that appear.
Once it is fully rolled, give it a few squeezes to make sure everything is tightly stuck together.
Wrap the roll in plastic wrap and place in the fridge for 30 min.
Preheat the oven to 350 F.
With a sharp knife, slice the roll into 1 cm cookies. If they get a little smushed gently reshape them into a circle. Place the cookies on baking sheets leaving 1" between each cookie.
Bake the cookies for 10 min. These cookies should stay relatively pale in color.
Allow the cookies to cool on the baking sheet for 5 min, then transfer the cookies to cooling racks and allow them to cool completely.
Icing
After the cookies have cooled fully (about 15-20 min). They are ready to be iced!
Whisk together the powdered sugar, vanilla extract, and milk. This icing should be in between a glaze and a thick icing - somewhere around maple syrup. If you need to thin it out, add more milk. If it needs to be thicker add more powdered sugar.
Dunk each cookie halfway into the glaze. Scrape off any extra from the bottom and side edge and then place on a piece of parchment or a wire cooling rack to allow the icing to harden.
Notes
These will last at least one week in an airtight container at room temp, be sure to let the icing dry fully before stacking them.