Bring the carrot juice, 3 cups water and the mushroom stems to a boil in a saucepan over medium heat. Meanwhile, heat the olive oil and 2 tablespoons butter in a pot over medium-high heat. Add the mushroom caps and barley and cook until the barley is toasted, about 5 minutes.
Add the onion, celery and rosemary to the pot and season with salt and pepper. Cook, stirring occasionally, until the onion is translucent, about 2 minutes. Add the carrots and cook 2 more minutes.
Increase the heat to high and add half of the carrot juice mixture, leaving the mushroom stems in the pan. Bring to a boil and cook, stirring occasionally, until the liquid is almost absorbed, about 6 minutes.
Add the remaining carrot juice mixture, the kale and ginger and cook, stirring, until the barley and vegetables are tender, about 5 minutes. Stir in the remaining 2 tablespoons butter, then ladle the stew into bowls.
Photograph by Antonis Achilleos
Courtesy of Food Network Magazine