In a large pot, melt butter over medium heat. Add diced onions and cook for 5 minutes, until soft and translucent.

Add chopped cabbage to the pot and cook until it wilts, about 5-7 minutes.

Add the chopped potatoes, vegetable stock, and black pepper to the pot. Bring to a simmer and cook for 20 minutes, or until the potatoes are tender.

Use an immersion blender to process the soup until it’s thick and slightly chunky. Alternatively, transfer it to a food processor in batches and blend until smooth.

Stir in milk or half & half, bring the soup back to a simmer, and adjust seasoning with salt and additional pepper if needed.

Ladle the soup into bowls and serve hot with a slice of crusty Irish soda bread or your favorite artisan loaf.