In a large soup pot or Dutch oven, cook the chopped bacon over medium heat until browned and crispy. Use a slotted spoon to remove the bacon and set aside, leaving the drippings in the pot.

Add butter, diced onion, and garlic to the pot. Sauté until the onion is soft and translucent, about 5 minutes.

Stir in chopped cabbage and thyme, and cook for 5 more minutes, until the cabbage begins to soften.

Add diced potatoes (and carrots if using), and pour in the broth. Bring to a boil, then reduce heat and simmer for 20–25 minutes, or until the potatoes are tender.

Stir in the milk or half-and-half, and return the cooked bacon to the pot. Simmer gently for another 5 minutes. Season with salt and pepper to taste.

For a creamier texture, use a potato masher to lightly mash some of the soup directly in the pot, or blend part of it with an immersion blender.

Ladle into bowls and garnish with fresh parsley and extra bacon bits if desired.