Preheat the oven to 350°F (175°C) and prepare a 9x13-inch baking dish.
In a small bowl, whisk together the all-purpose flour, baking soda, and salt until evenly combined.
Place the butter in a skillet over medium heat and cook until it foams and brown bits form on the bottom; once the bits appear, turn off the heat and stir for about 30 seconds, then let the brown butter cool slightly.
Brush a little of the warm brown butter over the bottom and sides of the 9x13-inch dish to grease it.
In a large bowl, place the light brown sugar and pour in the remaining brown butter; mix until smooth and partially dissolved into the sugar.
Add the eggs one at a time, stirring after each addition so the eggs don’t cook if the butter is still warm, then stir in the vanilla extract.
Using a wooden spoon, stir the dry flour mixture into the wet ingredients until almost combined, then fold in the crushed Oreo pieces until evenly dispersed.
Press the mixture evenly into the prepared baking dish and bake for 25 to 30 minutes, until the top is golden brown and set in the center.
Remove from the oven and, if desired, sprinkle with flaky sea salt; allow the bars to cool completely in the pan before slicing into 16 bars.