Heat oil in a skillet over medium heat; stir in onion. Cook and stir onion until golden, about 5 minutes. Add mushrooms and cook until softened, about 5 minutes more. Pour in 1/2 cup broth; simmer, stirring occasionally, until liquid is reduced, 8 to 10 minutes.
Whisk miso paste and 2 tablespoons broth together in a bowl. Pour into the vegetable mixture. Season with salt and pepper. Cook and stir to coat the vegetables, about 1 minute. Add chicken meat and the remaining broth. Reduce heat to low and simmer soup until flavors are well blended, about 30 minutes.