Preheat oven to 375.
In a heavy saucepan melt the butter. Continue cooking butter until it is bubbly and you begin to see browned bits on the bottom. The bubbling will subside a bit when it is ready. Then make sure the heat is on low. When the butter is browned, add the brown sugar and salt, whisk about two minutes to combine, it will not be a smooth mixture, that is ok. Pull from the heat and immediately and SLOWLY whisk in the heavy cream and continue whisking until fully incorporated.
Let it sit for about 20 minutes or so off the stovetop and then add your eggs one at a time, whisking like crazy after adding each egg. Add the vanilla last.
Pour the butterscotch mixture into the prebaked pie crust and bake at 375 for 10 minutes ONLY! Then turn the oven down to 300 (this will ensure a smooth and not dry custard) and continue to bake for 30 more minutes - maybe a touch more. It will be a little jiggly in the center, but only barely.
Cool this completely before serving. I am not a fridge fan for this pie (nor am I for pumpkin pie), but my family def prefers this cold, so do what you like. Definitley doesn't hurt to serve with unsweetened (that's right, the pie is sweet enough) whipped cream on the top.