In a stock pot or sauté pan, cook 2 sticks of butter until golden brown and bubbly. Turn off heat and strain through a fine-mesh sieve into a heat-proof bowl and set aside.
In another large bowl, mix flour, cornstarch, salt, and baking powder.
Once brown butter has come to room temperature and regained a softened butter texture, beat with a hand mixer with granulated sugar and 2 tbsp powdered sugar until light and fluffy.
Add your eggs, one at a time, and vanilla extract, until a smooth batter forms. Then, add your dry ingredients and mix until a stiff dough forms.
Split dough in half and bring each into a cohesive disc. Wrap each with plastic wrap and place in fridge to chill for at least 2 hours.
When ready to bake, preheat oven to 350°F. Remove dough discs from fridge and let come to room temperature, for about 15-30 minutes.
Line two 8x8" baking pans with parchment paper. Take each dough and press into the bottom of each pan until you fill the entire bottom evenly.
Chill each pan of dough for 10-15 minutes before baking. Then, bake for 12-15 minutes until lightly golden brown.
Once done, immediately remove each cookie square from the pans using the parchment paper as a handle. Trim off the edges (about 1/4") and cut each large square into 4 equal 4x4" squares using a ruler and knife or pizza cutter.
In a large bowl or stand mixer, beat 3 2/3 cup powdered sugar with 3 egg whites, salt, and lemon zest until smooth, being careful not to splatter your sugar.
Beat on high speed until a stiff meringue forms, about 5 minutes. Remove 1/4 of your icing and transfer it to a piping bag to use as your border icing. Then, add your lemon juice to the rest of your icing.
Separate your remaining icing into two bowls, one with 3/4 of the total and one with 1/4 of the total. Add blue food coloring to the smaller amount until you achieve a saturated phthalo hue. Cover and reserve for decorating.
Transfer your white flood icing to an another piping bag and label with "flood" or "thin".
Once your cookies are cool, flip squares over onto the flat side. Start with one square and pipe a border of your stiff icing around the outer edge. Then, flood the inside of the cookie completely with your white flood icing, using a toothpick to fill in gaps. Let dry completely, for at least 1-2 hours or up to a day.
Once your white icing has hardened completely, decorate your tiles with your blue icing. Mix with more lemon juice or water to thin into a desired paint consistency and use a clean paintbrush to apply designs of your choice. Let dry and serve.