In an electric mixer fitted with the paddle attachment, mix the butter and sugar together until just incorporated. Do not over-mix at this stage, or the cookies may spread while baking. Add the egg and vanilla extract. Mix again on low speed, stopping to scrape down the sides of the bowl intermittently as needed.
In a medium bowl, whisk together the flour and salt. Add to the butter and egg mixture. Mix on low speed until a dough has formed and there are no longer any streaks of butter in the mixing bowl. (The dough will often clump around the paddle attachment while being mixed; this is normal and a good sign that the dough is the right consistency.) If your mixture does not come together and is crumbly, add ice-cold water, one tablespoon at a time, until the dough clumps. The dough should be firm and malleable and easily rolled into a ball between your palms without sticking to your hands. If the dough is too sticky, add additional flour 1 tablespoon at a time.
Roll the dough into a disk and cut away 1/3 of the dough. Return this 1/3 portion of dough to the mixer bowl and add the cocoa. Mix until well incorporated. Cover the brown dough with plastic wrap while you shape the rest of the dough. Roll the remainder of the dough into an 8-inch baton. Using an index finger, make an indention along the top center of the dough. Make indentions in the same way on both sides of the dough baton, about 1/2 inch from the top. This creates a bread top (crown) shape. Loosely wrap the baton in wax paper and transfer to the freezer. Chill until very firm, about 15 minutes.
Meanwhile, roll the chocolate dough to 1/4-inch thickness between two sheets of parchment paper. When the plain dough is well-chilled, lightly brush the chocolate dough with egg wash and place the chilled dough on top of the chocolate dough; wrap a single layer of chocolate dough around it. Smooth the chocolate dough around the plain dough using your fingers to train it into the grooves and contours until there are no air bubbles between the two doughs. Cut away the excess chocolate dough and reserve for other use (or cut into shapes and bake).
Wrap the cookie dough loosely in parchment or wax paper and transfer to the freezer until partially frozen, about 15 minutes.
Preheat the oven to 350°F.
When the dough is chilled, slice the baton into 1/4 inch cookies with a large, non-serrated knife. Place the slices on parchment-lined cookie sheets and bake for 10-12 minutes, or until the cookies are fragrant and the surface has a matte appearance. Transfer the cookies to a wire rack to cool completely.
Combine the butter and powdered sugar in an electric mixer fitted with the whip attachment. Add the powdered sugar and mix on low speed until combined. Mix in the extract. Add milk or cream as needed to bring to spreading consistency. Divide the frosting into two bowls and tint each with the food color; mix well until no streaks of white remain. Add a little of the yellow food color to the bowl tinted moss green, which will make it more similar to avocado color.
Spread the frosting onto the centers of some of the cookies. Leave others plain for fried egg decors, if you’re making them.
Use a small fork to make tine marks in the avocado green buttercream; sprinkle on coarse crystal sugar for ‘salt.
Spread yellow buttercream over toast, or quarter starburst candies and use as ‘butter pats’ affix them to the cookies with yellow buttercream.
Spoon small amounts of melted white chocolate into irregular oval shapes onto a parchment-lined baking sheet. Immediately top each with one or two yellow Sixlets candies. Let firm and then affix them to cookies with a dot of yellow buttercream. Pepper can be created from the chocolate cookie portion of an Oreo, crush it fine between your fingers and sprinkle over the egg. (You can also top the avocado toasts with an egg, which is really cute!)
Spread cookies with yellow buttercream and top with rainbow nonpareils.
Sprinkle a little ground cinnamon over a plain cookie and mottle with fingers. Cookies can be served whole or sliced into toast points.
Keyword kid friendly, slice and bake, sugar cookies