1. In an 8- or 9-inch-wide pan, mix shallots with oil. Bake in a 375° oven until deep golden, shaking pan occasionally to turn pieces, about 1 hour.

2. Meanwhile, in a 5- to 6-quart pan over medium-high heat, frequently stir onions, butter, and sugar until onions are deep golden brown, about 45 minutes.

3. Add sherry to shallots, stirring to release browned bits in pan. Pour mixture into onions. In a blender, whirl vegetables, a portion at a time, until very smooth, adding enough broth to facilitate blending. Return puréed vegetables to pan. Add remaining broth, thyme, and sage.

4. Stirring often, bring to a boil over high heat, then reduce heat and simmer 20 minutes to blend flavors. Add half-and-half and stir over medium-high heat until hot, 2 to 3 minutes. Add parsley.

5. Ladle into bowls and season to taste with hot sauce, salt and pepper, and juice from lemon wedges.