Preheat the oven to 375°. Using a mandoline, slice the potatoes and squash lengthwise 1/8 inch thick. Slice the rutabaga crosswise 1/8 inch thick.
Spray an 8-by-12-inch glass baking dish with cooking spray. Arrange half of the potatoes in the dish, overlapping them slightly; season with salt and pepper. Top with half of the rutabaga and the squash, seasoning each layer. Repeat the layering. Pour the broth over and around the vegetables.
Cover tightly with foil and bake for 1 hour, until the vegetables are almost tender when pierced. Remove the foil and pour the cream over the gratin. Bake for about 30 minutes longer, until the liquid has thickened.
Preheat the broiler. Mix the panko with the oil and season with salt and pepper; sprinkle over the gratin. Broil 3 inches from the heat for 2 minutes, until golden, rotating for even browning. Let the gratin stand for 10 minutes, then serve.