Position a rack in the center of the oven and preheat to 300℉. Coat an 8 x 8-inch baking pan with cooking spray or brush with butter. Line the pan with two crossed strips of parchment paper that hang over the sides.
Melt the butter in a medium saucepan over medium heat, stirring occasionally, until it foams and the milk solids turn deep golden brown and smell nutty, 5–8 minutes. Remove from the heat to stop the cooking and let the browned butter cool in the pot until just warm before using.
In a large bowl, whisk together the flour, powdered sugar, and salt until evenly combined and no lumps remain; sift to remove any stubborn lumps if needed.
Add the browned butter and stir with a spatula until just combined and no streaks of flour remain.
Transfer the shortbread dough to the prepared pan and press it into an even layer with your hands or a flat-bottomed measuring cup.
Sprinkle about ¼ cup of toffee bits evenly over the surface of the dough.
Bake the shortbread until 5–8 minutes shy of doneness, about 40 minutes. Working quickly, sprinkle a generous layer of toffee bits over the surface and lightly press them in with a flat-bottomed measuring cup (use oven mitts). Return to the oven and bake for another 5–8 minutes, until the edges are lightly golden and crisp, 45–50 minutes total.
Let the shortbread cool completely in the pan. If desired, sprinkle a pinch of flaky sea salt over the top while it is still warm. Once cool, use the parchment overhang to lift the shortbread out of the pan, then cut into squares or rectangles, or break into pieces by hand.
