In the bowl of a stand mixer, combine the powdered sugar, butter, sugar and sea salt. With the paddle attachment, beat the mixture for at least 5 minutes, until it’s well combined. Add the egg and vanilla and beat for at least 3 minutes to combine. The mixture will look clumpy and wet.
Place a sieve over the bowl and sift in the flour and cocoa. Cover the bowl with a tea towel and beat on medium low until the dough comes together. Use your hands and gather the dough into a flat disk. Wrap it in plastic wrap and set in the fridge for two hours until the dough is firm but not too stiff to roll out. If you’ve left it for a few hours or overnight, let it soften on the counter until it’s workable
On a lightly floured surface, roll out the dough to 1cm thick and use a cookie cutter to cut out shapes. Set the shapes on a parchment lined cookie sheet. Don’t crowd them but also don’t worry about leaving too much space, they won’t spread. Set the tray in the freezer while the oven preheats to 350 F.
Bake for 10 minutes. Because they are dark you won’t be able to see any browning, we want these slightly underbaked anyway so they’ll be soft inside.
Make the frosting: Beat the butter, vanilla and salt together. Add the powdered sugar and beat until smooth. Stir in blue food coloring.
Frost the cookies. Then to get the speckled look, mix a bit of cocoa with water (1 tablespoon of water and a 1/2 tsp cocoa). Dip a paint brush in the mix then tap it over the cookies. It will splatter all over.