Melt butter in 6-quart pot over medium-high. Cook mushrooms 8 to 10 minutes until loose water evaporates and edges brown.
Cook onion and carrot 4 minutes. Add barley, broth, thyme, salt, and pepper; simmer.
Loosely cover; simmer over medium-low 30 to 40 minutes, stirring occasionally, until barley chewy-tender. Follow longer package time if needed.
Add turkey; gently simmer 5 to 8 minutes to 165°F. Whisk miso smooth with 1/2 cup hot broth in bowl.
Remove from heat; add miso mixture. Serve while broth stays loose around barley.