Preheat oven to 300°F. Melt butter in a small skillet over medium heat and cook, stirring often, until milk solids begin to brown, about 6 minutes (do not burn). Strain through a fine-mesh sieve into a medium bowl. Chill until just firm, about 30 minutes.
Using an electric mixer on medium-high speed, beat brown butter and powdered sugar in a large bowl until pale and creamy, about 2 minutes. Reduce speed to low and gradually beat in flour and salt, scraping down sides of bowl as needed, just to blend (do not overwork dough).
Transfer dough to a sheet of parchment. Form dough into a rectangle. Place second sheet of parchment over dough. Roll out dough to a 8×6″ rectangle about 1/2″ thick.
Remove top sheet of parchment. Transfer dough on parchment to a rimmed baking sheet. Using a fork, pierce dough all over, spacing 1″ apart. Bake shortbread, rotating halfway through, until golden brown and firm to the touch, 55–65 minutes.
Cut shortbread lengthwise into 1″ strips on baking sheet while still warm. Cut each strip crosswise into thirds. Let cool completely.