Position a rack in the center of the oven and preheat to 350 degrees . Unroll the piecrust into a 9-inch pie pan; crimp the edge and prick the bottom all over with a fork. Cover the crust with a sheet of foil and fill it with dried beans or other pie weights. Bake until the crust edge is just dry and lightly colored, about 10 minutes. Remove the foil and beans.
Meanwhile, pierce the sweet potatoes in several places. Microwave on a paper towel, turning once, until they are easily pierced with a fork, about 7 minutes. Let cool, then peel. Using a food processor, pulse the sweet potatoes until smooth. Blend in the buttermilk, then add the eggs, brown sugar and salt and pulse until combined, about 30 seconds. Pour the filling into the prepared crust. Bake the pie until the center is just set and puffed and the pastry is golden, about 50 minutes.
Preheat the broiler. Cover the top of the pie with the large marshmallows, mounding them in the middle; fill the empty spaces with the mini marshmallows. Place the pie on a baking sheet and broil until browned, less than a minute. Turn off the broiler, close the oven door and leave the pie in the oven until the marshmallow topping just softens, about 3 minutes. Transfer to a rack to cool slightly.