Combine carrot juice, 3 cups water, and mushroom stems in a saucepan; bring to a boil over medium heat.

Heat olive oil and 2 tablespoons butter in a pot over medium-high heat.

Add mushroom caps and barley to the pot and cook until barley is toasted, about 5 minutes.

Stir in onion, celery, and rosemary; season with salt and pepper.

Cook, stirring occasionally, until onion is translucent, about 2 minutes.

Stir in carrots and cook 2 more minutes.

Increase heat to high and add half of the carrot juice mixture, leaving mushroom stems in the pan.

Bring to a boil and cook, stirring occasionally, until the liquid is almost absorbed, about 6 minutes.

Add remaining carrot juice mixture, kale, and ginger; cook, stirring, until barley and vegetables are tender, about 5 minutes.

Add remaining 2 tablespoons butter, then ladle the stew into bowls.

Serve warm.