Melt 2 tablespoons butter in each of 2 large skillets over moderate heat. Divide onions, shallots, and 1 tablespoon thyme between pans and cook, stirring occasionally, until onions are golden brown (do not burn), about 1 hour. Add 1/2 cup stock to each skillet and simmer, stirring occasionally, until liquid reduces to a glaze. Transfer both onion mixtures to a 5-quart pot; stir in 4 cups water and remaining 3 cups stock. Simmer soup, uncovered, until reduced to 6 cups. Add sherry; salt and pepper to taste. Remove from heat; keep soup hot, covered.

Meanwhile, stir together goat cheese, 2 tablespoons oil and remaining 2 teaspoons thyme; salt and pepper to taste. Preheat broiler. Brush bread slices on one side with remaining 1 tablespoon oil and broil, turning, until golden brown. Spread toasts with goat cheese mixture. Serve soup ladled into bowl with cheese toasts on side. Makes 6 servings.