In a large frying pan over medium-high heat, melt butter. Add shallots and sprinkle with salt. Cook, stirring occasionally, until shallots are well browned all over, about 20 minutes.
Add walnuts, Vin Santo, and thyme. Bring to a boil, then reduce heat to low, cover, and simmer until shallots are tender, about 5 minutes. Uncover, turn heat to medium-high, and cook until liquid is reduced and almost completely evaporated. Serve warm.
* Vin Santo is sold at liquor stores; substitute Madeira, Marsala, Port, or any other sweet wine, if you prefer.
Note: Nutritional analysis is per serving.