Preheat oven to 400 degrees F. Coat a 9-by-13-inch baking dish with cooking spray.
If using parsnips, quarter lengthwise and remove the woody core before cutting into 1/8-inch thick slices. Cook vegetables in a large pot of boiling water until barely tender, about 5 minutes. Drain.
Heat 2 tablespoons oil in a medium saucepan over medium heat. Add shallots and cook, stirring occasionally, until light brown, 3 to 4 minutes. Add 1 cup milk and bring to a simmer. Combine flour and the remaining 1/3 cup milk in a small bowl to make a smooth paste; stir into the hot milk and cook, whisking constantly, until the sauce bubbles and thickens, 1 to 2 minutes. Remove from the heat. Stir in 3/4 cup cheese, thyme, salt and pepper.
Combine breadcrumbs, the remaining 3/4 cup cheese and 1 tablespoon oil in a bowl.
Layer the vegetable slices in the prepared baking dish. Pour the cheese sauce over the top and top with the breadcrumb mixture.
Bake the gratin until it is bubbling and the top is golden, 30 to 40 minutes. Let cool for 10 minutes before serving.
Tips: Beets, carrots and parsnips are easily peeled with a vegetable peeler, but for tougher-skinned roots like celeriac, rutabaga and turnips, removing the peel with a knife can be easier. Cut off one end of the root to create a flat surface to keep it steady on the cutting board. Follow the contour of the vegetable with your knife. If you use a vegetable peeler on the tougher roots, peel around each vegetable at least three times to ensure all the fibrous skin has been removed.
To make fresh breadcrumbs, trim crusts from country-style whole-wheat bread. Tear bread into pieces and process in a food processor until coarse crumbs form. One slice of bread makes about 1/2 cup crumbs.
Per serving: 276 calories; 13 g fat (5 g sat, 6 g mono); 25 mg cholesterol; 30 g carbohydrates; 12 g protein; 6 g fiber; 357 mg sodium; 604 mg potassium.