Preheat the oven to 450 degrees F.
In a small saucepan over medium heat, melt the butter. Stir in the brown sugar. When melted, remove from the heat. Stir in the flour and nutmeg.
Pile the apricots into the prepared piecrust. Spread the butter mixture on top, patting it all the way to the edges.
Bake the pie in the bottom third of the oven for 10 minutes. Reduce the temperature to 350 degrees F. and bake 45 to 50 minutes longer, until the topping is golden brown and the apricots are very tender. Cover loosely with foil if the top is getting too brown. Let cool on a rack.
Recipe courtesy of Joel Patraker and Joan Schwartz, The Greenmarket Cookbook,Penguin Putnam Inc. 2000