Position a rack in the center of the oven and heat the oven to 400°F.
Bring a 4-quart pot of well-salted water to a boil over high heat. Add the rutabaga and cook until just tender, about 6 minutes. Drain well and spread out on a baking sheet to cool at room temperature.
Coat an 11-inch (1-1/2-quart capacity) oval gratin dish (or other small gratin dish) with about ½ Tbs. of the butter. In a 10-inch skillet, melt the remaining butter over medium heat. Add the prosciutto and cook, stirring occasionally, until just beginning brown to and crisp, about 2 minutes. Off the heat, toss the breadcrumbs and garlic with the prosciutto and butter. Set aside.
Arrange a single layer of rutabaga slices in the gratin dish, slightly overlapping. Season evenly with salt and pepper and sprinkle half of the Gruyère and thyme on top. Repeat with another layer of rutabaga, salt, pepper and the remaining thyme and Gruyère. Top with a final layer of rutabaga and season again with salt and pepper. (You can prepare the recipe to this point up to 2 hours ahead. Cover and keep at room temperature.)
Bring the cream to a simmer in a small saucepan. Gently pour the cream over the rutabaga and scatter the prosciutto mixture on top. Bake until the rutabaga can be easily pierced with a fork, and the prosciutto topping is deeply golden, 25 to 30 minutes. Let sit for about 10 minutes before serving.