Heat the oven to 400°F. Put the onions and 3/4 cup water in a large saucepan over high heat. Stir and separate the onions with a fork as they heat. When the water boils, reduce the heat to medium, cover, and simmer for 5 minutes. Drain well and pat dry.
Combine the cream, thyme, and 1/2 tsp. salt in a small saucepan over medium-high heat. When the cream comes to a boil, reduce the heat to a simmer and cook for 5 minutes, stirring occasionally.
Meanwhile, brush a shallow 2-qt. gratin or baking dish with 1 Tbs. butter. In a small bowl, toss the breadcrumbs, Parmigiano, savory, the remaining 2 Tbs. melted butter, 1/2 tsp. salt, and several grinds of pepper.
Spread the onions in the baking dish. Pick out and discard the thyme sprigs from the cream. Pour the cream over the onions and scatter the breadcrumbs on top. Bake until the breadcrumbs are a deep golden brown and the cream is bubbling furiously around the edges, about 30 minutes. Let rest 10 minutes before serving.