For fresh mushrooms, brush clean, trim ends, and thickly slice. For dried, soak in hot water to cover, swishing occasionally, until pliable, 30 to 60 minutes; lift mushrooms out, squeezing out liquid.
Meanwhile, rinse chard well. Tear leaves from stems into large pieces. Very thinly slice stems on a diagonal and set aside, separate from leaves.
In an 8- to 10-qt. pot over medium heat, toast walnuts in 1 tbsp. butter, stirring often, until golden, 5 to 8 minutes. With a slotted spoon, transfer nuts to a bowl.
Add mushrooms, 2 tbsp. butter, and 1/4 tsp. each salt and pepper to pot; sauté until mushrooms are softened, 5 minutes. Transfer to a bowl. Increase heat to medium-high. Sauté stems from chard and onion in pot with remaining 1 tbsp. butter and 1/4 tsp. each salt and pepper until onion is golden, about 15 minutes.
Stir in half of chard leaves, 1/4 cup water, and 1/4 tsp. each salt and pepper. Cover and cook, stirring often, just until wilted, 2 minutes. Add remaining chard and 1/4 tsp. each salt and pepper; cook, covered, until wilted, 2 to 3 minutes more. Stir in mushrooms and half of nuts; transfer with a slotted spoon to a shallow bowl. Sprinkle with remaining nuts.