Preheat oven to 350°F. Thoroughly grease a 9.5-inch pie plate with butter.
Place the eggs and sugar in a large, deep mixing bowl. Using a hand mixer on medium speed, beat for 1 to 2 minutes, until the mixture is pale yellow, thickened, and slightly fluffy.
Put the hand mixer away. Stir in the melted butter and vanilla until combined. Gradually stir in the flour and salt until incorporated. Slowly add the milk, stirring until combined. The batter will be very thin and watery; this is normal. Gently stir in the coconut.
Give the batter one final stir from the bottom of the bowl, then pour it into the prepared pie plate. Bake for 45 to 50 minutes, until the top is golden brown and the center is mostly set.
Place on a wire rack and let cool for at least 2 hours. The pie may puff slightly as it bakes and then settle as it cools, forming its own crust and layers. For the best texture, refrigerate for at least 1 hour before serving to allow the custardy center to firm up. If you prefer, you can also serve the pie warm. Garnish with powdered sugar and whipped cream, if desired.
Store covered and refrigerate leftovers for up to 4 days.