How to Make It
Preheat the oven to 400°. Spread the carrots and beets on 2 separate sheets of foil and drizzle with 2 tablespoons of the olive oil. Fold up the edges of the foil sheets to make two sealed packets. On a small baking sheet, rub the fingerling and sweet potatoes with olive oil and prick them all over with a fork. In a medium baking dish, toss the cauliflower and broccoli florets with the remaining 1 tablespoon of olive oil. Roast the broccoli and cauliflower for about 15 minutes, the potatoes for about 30 minutes and the carrots and beets for about 35 minutes, until all of the vegetables are tender and the broccoli and cauliflower are browned in spots. Let cool slightly. Peel the sweet potatoes and cut them into quarters lengthwise; peel the fingerling potatoes and cut them in half lengthwise. Season the roasted vegetables lightly with salt and pepper.
In a blender, combine the half-and-half with the eggs, flour, 1/2 cup of the cheese and 2 teaspoons salt and blend until smooth.
Butter a 9-inch-square glass or ceramic baking dish and sprinkle all over with cheese. Arrange the vegetables in the baking dish in 2 stacked layers, lining them up in the same direction. Pour the batter on top. Sprinkle with the remaining 2 tablespoons of cheese and bake in the center of the oven for about 45 minutes, until puffed, golden and set; cover the dish with foil halfway through to prevent over-browning. Let the clafoutis cool slightly, then cut into squares and serve.