In a large pan over medium heat, heat the oil until shimmering. Saute the garlic, carrot, tomato paste, and thyme until the tomato paste starts to darken, 2-3 minutes.
Add the mushrooms and soy sauce. Cook until the mushrooms start to brown, 3-5 minutes. Remove and discard the thyme stem. Stir in the barley and heat through, about 1 minute.