Heat your oven: Preheat to 300°F and line an 8-inch square cake pan with parchment paper.
Chill the browned butter: Freeze butter in a bowl for 30 minutes, stir, then return to the freezer for another 30 minutes until solid. Cut into chunks.
Brown your butter: In a medium saucepan or frying pan, melt butter over medium-high heat until it turns clear golden and toasted speckles appear. Remove from heat and transfer to a medium bowl then freeze until solid.
Make the shortbread dough via stand mixer or food processor: In a stand mixer, beat browned butter chunks, brown sugar, and salt together until mixed. Add vanilla, water, and flour, and mix until combined. In a food processor, pulse brown sugar, salt, flour, and then add butter, vanilla, and water. Run until smooth.
Transfer dough to pan: Press shortbread dough into prepared pan and smooth the top with an offset spatula.
Bake the shortbread: Bake for 30 minutes. Remove from the oven to shape the shortbread.
Shape the shortbread: Lightly score the surface with a paring knife or bench scraper to create diamond shapes, then cut along these lines with a paring knife.
Finish baking: Return to the oven for another 15-20 minutes until golden at the edges. Let cool in the pan, then separate the shortbread pieces.
Optional icing: Mix powdered sugar and lemon juice to a thick but drizzle-able consistency. Drizzle over shortbread and let set for an hour.
Storage: Keep shortbread at room temperature for 1 week, or freeze and ice after defrosting.