Butter a shallow 1 1/2-quart baking dish or a 10-inch round or oval gratin dish and set aside.
In a large mixing bowl, toss together carrots, turnips, scallion greens, parsley and 3 tablspoons of cornstarch.
Spread vegetable mixture in prepared pan, pressing and smoothing it into place.
In a small saucepan, dissolve remaiing cornstarch in 1/4 cup of the milk.
When completely dissolved, add cream and remaining milk.
Bring to a boil over moderately high heat, whisking constantly.
Reduce to moderate heat.
In a medium mixing bowl, beat egg with a fork and season with salt and pepper.
Add egg to milk mixture in slow, steady stream, beating until combined.
Pour this hot custard over the vegetable mixture in gratin dish.
Sprinkle top evenly with Parmesan cheese.
(Gratin can be prepared 24 hours ahead to this point and refrigerated.) Dot gratin with butter and make in the middle of a preheated 375F oven for 45 minutes, or until it is bubbling through a golden crust.
Let gratin stand 10 minutes before serving to crisp crust and allow custard to set.