Brown the butter in a medium saucepan. Pour butter and all the toasty flecks into a medium bowl when done and move to the freezer

After 30 minutes, stir the butter, scraping down the sides, and mixx the melted parts over the solidified chunks. Return to the freezer for another 30 minutes

Cut the frozen butter into chunks

Make the shortbread dough one of two ways:

In a stand mixer with the paddle attachment, beat browned butter chunks, brown sugar, and salt together, scraping frequently, until evenly mixed

Add vanilla, water, and flour, and mix just until combined, scraping down the bowl again

Bake the shortbread: For 30 minutes — it will not be fully baked yet — and remove from the oven just long enough to shape the shortbread

Use the back of a paring knife to make light line indentations barely 1/4-inch into the shortbread, about 1 inch apart. Then, at about a 45-degree angle from the first lines, make remaining indentations 1.5 inches apart

Use a very thin, sharp paring knife to cut along these lines to the bottom of the pan. Use the back of a wooden skewer to dock the shortbread about 1/3 of the way into the cookie

Return the shortbread to the oven for another 15 to 20 minutes, or until more deeply golden at the edges. Let cool in the pan. Once cooled completely, separate the shortbread where they were cut. If needed, use your paring knife again to make clean separations

Space shortbread evenly out on a rack and place over a tray to catch drips

Place powdered sugar in a small bowl and add 1 tablespoon lemon juice, mixing to combine

Transfer icing to a sandwich bag and snip a tiny bit of the corner off and drizzle the icing over the shortbread. Let set for one hour at room temperature