Preheat your oven to 375°F (190°C) and place the unbaked pie crust into a standard 9-inch pie dish.

In a small saucepan over medium heat, melt the butter and cook until it foams and turns a golden amber color with a nutty aroma (about 5-7 minutes). Watch closely to prevent burning, then remove from heat.

In a medium mixing bowl, whisk together the packed brown sugar and all-purpose flour until there are no lumps.

Pour the hot browned butter, heavy cream, maple syrup, and vanilla extract into the sugar mixture, whisking vigorously until the filling is smooth and fully combined.

Pour the filling directly into the prepared pie crust.

Bake for 35 to 40 minutes, or until the crust is golden brown and the filling is bubbling but still slightly jiggly in the center.

Remove from the oven and immediately sprinkle the flaky sea salt over the top.

Allow the pie to cool completely at room temperature for at least 2 hours to let the filling set before slicing and serving.