This rich side dish could easily stand on its own as an elegant meatless main course.
Melt 3 tablespoons butter in large deep nonstick skillet over medium-high heat. Add shallots and sauté until tender and golden, about 6 minutes. Add garlic and stir 30 seconds. Add remaining 3 tablespoons butter and stir until melted. Add mushrooms, sprinkle with salt and pepper.
Sauté until tender and brown, about 10 minutes. Add thyme and stir 1 minute. Add Madeira and simmer until almost evaporated, about 1 minute.
Add chestnuts and whipping cream and simmer until cream thickens and coats mushroom mixture, about 1 minute. Season generously with salt and pepper. Transfer to bowl; sprinkle with chives.