In a food processor or bowl of an electric mixer, cream together the brown butter, powdered sugar, and salt until smooth. Slowly add in the flour until everything is combined, scraping down the sides as needed. Fold in the toffee pieces with a rubber spatula.
On a lightly floured surface, shape the dough into an 8-10 inch log. Wrap it in plastic wrap and refrigerate for one hour.
Place the racks in the upper and lower third of the oven. Preheat the oven to 350°F.
Slice the shortbread cookies into ¼" pieces and roll the edges in the turbinado sugar. Transfer them to two parchment paper lined baking sheets with 12-14 cookies on each pan. Bake for 16-18 minutes, rotating the pans halfway through. Let the shortbread cookies cool for 5 minutes before transferring to a cooling rack.