Preheat your oven to 375°F (190°C). Place the unbaked pie crust in a 9-inch pie dish and crimp the edges to your liking.

In a small saucepan, melt the butter over medium heat, swirling occasionally. Cook until the butter foams and brown specks appear at the bottom, emitting a nutty aroma, about 3 to 5 minutes, then remove from heat to cool slightly.

In a medium mixing bowl, whisk together the packed light brown sugar and all-purpose flour until no lumps remain. This ensures the filling will set properly without graininess.

Pour the cooled brown butter, heavy cream, egg, vanilla extract, and salt into the sugar mixture. Whisk everything together until the batter is completely smooth and cohesive.

Carefully pour the filling mixture into the prepared pie crust, spreading it out evenly. Place the pie dish on a baking sheet to catch any potential spills.

Bake for 40 to 45 minutes, or until the edges of the filling are set and slightly puffed, but the center still has a slight jiggle. If the crust edges brown too quickly, cover them with foil halfway through baking.

Remove the pie from the oven and let it cool completely on a wire rack for at least 2 hours before slicing. Serve warm or at room temperature, ideally with a dollop of whipped cream or vanilla bean ice cream.