Swiss chard lends a pleasant earthiness to this hearty stuffing; meaty chestnuts add texture and richness. Choose high-quality crusty bread for this recipe to ensure a fluffy texture and crisp top.
Combine vegetable broth and mushrooms in a medium saucepan; bring to a boil over high. Reduce heat to medium-low; simmer, undisturbed, until reduced to about 2 1/2 cups, about 30 minutes. Pour through a fine wire-mesh strainer into a heatproof bowl; discard solids. Add water as needed to measure 2 1/2 cups.
Preheat oven to 375°F. Grease a 3- to 4-quart baking dish with butter. Heat butter and oil in a large skillet over medium until butter is melted. Add onion, celery, Swiss chard stems, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper. Cook, stirring occasionally, until vegetables soften and just start to brown, about 10 minutes. Add sage; cook, stirring occasionally, 1 minute. Gradually fold in Swiss chard leaves until just wilted. Remove from heat.
Whisk together 2 1/2 cups broth, eggs, remaining 2 teaspoons salt, and remaining 1 teaspoon pepper in a large bowl. Add chard mixture, toasted bread, chestnuts, and parsley; stir well to combine. Spread mixture evenly in prepared baking dish. Cover tightly with aluminum foil; let stand at room temperature 45 minutes.
Bake in preheated oven until heated through, about 30 minutes. Remove and discard foil; bake until lightly browned, about 30 minutes. Let stand 10 minutes.
Broth can be made up to 3 days ahead and stored in an airtight container in refrigerator.