Preheat the oven to 350 degrees F. The rack should be in the lower third of the oven.
In a microwave-safe bowl, melt the stick of butter using a low setting such as 30 or 40% power. You want the butter to be soft and goopy rather than liquid. Let it cool slightly. If you melt it at a low heat it won’t be too hot to begin with.
In another bowl, beat the eggs with a handheld electric mixer.
In a mixing bowl, combine the goopy, soft butter with the sugar and beat with the mixer until light. Add the beaten eggs and beat for a couple of seconds or just until mixed. Beat in the vanilla and salt.
With a large mixing spoon or silicone scraper, stir in the flour. When flour is incorporated, very gradually add the buttermilk, stirring until blended.
Pour into the pie shell and bake for 45 to 50 minutes or until top is lightly browned and center is slightly jiggly. If you use a ceramic pie dish such as an Emile Henry, you might need to add an extra 10 to 15 minutes (60 to 65). If pie is looking very brown toward the end but is not quite set, cover with a sheet of foil.
When the pie is done, it will not be completely stiff, but it should not be too loose and shaky, either. It should be just slightly jiggly while hot and should set as it cools.