Heat oven to 400 degrees. Combine cheeses in a large bowl. Transfer 1/3 of cheese mixture to a separate small bowl and set aside. Add cream, garlic and thyme to the large bowl with cheeses. Season generously with salt and pepper.

Grease a 2-quart casserole dish with butter. Working in batches, pick up a stack of each type of vegetable about 2-3 inches thick and slices and toss in the cream mixture until every slice is well coated. Neatly stack and lay in the casserole dish with the edges aligned vertically. Continue placing vegetables in the dish, working around the perimeter and into the center until the dish is packed tightly. Pour remaining cream mixture evenly over the vegetables just until the mixture comes halfway up the sides of the casserole, and discard any that remains.

Cover dish tightly with foil and transfer to the oven. Bake for 30 minutes. Remove foil and continue baking until the top is pale golden brown, about 30 minutes longer. Remove from oven, sprinkle with remaining cheese, and return to oven. Bake until deep golden brown and crisp on top, about 30 minutes longer. Remove from oven, let rest for a few minutes, top with fresh thyme and serve.