Melt butter in a 6-quart pot over medium-high heat. Cook mushrooms for 8 to 10 minutes until their loose water evaporates and edges brown.
Add onion and carrot and cook for 4 minutes. Stir in barley, broth, thyme, salt and pepper and bring to a simmer.
Cover loosely and simmer over medium-low heat for 30 to 40 minutes until barley is chewy-tender, stirring occasionally; follow a longer package time if needed.
Add roast turkey and simmer gently for 5 to 8 minutes until it reaches 165°F. Spoon 1/2 cup hot broth into a bowl and whisk miso into it until smooth.
Remove soup from heat, stir in the miso mixture and serve while the broth remains loose around the barley.