In a large saucepan, melt 1 piece butter over medium-high heat. Add one-third of the mushrooms and cook, stirring occasionally, until browned, 5 minutes. Season with salt and pepper and transfer to a bowl. Repeat in 2 batches with 2 more pieces butter and the remaining mushrooms.
Add the remaining 1 piece butter, the onion and all but 2 tablespoons of the scallions to the pot and cook, stirring, until golden-brown and tender, about 7 minutes. Meanwhile, thinly slice the potato. Return the mushrooms to the pot, add the potato, vegetable broth and 1 cup water and cook for 10 minutes.
Using a blender, puree the soup and transfer to a saucepan. Stir in the milk and 1/2 cup water and season with salt. Heat through over medium heat. To serve, top with the yogurt and reserved scallions.