Heat olive oil in a large pot over medium heat, then brown the kielbasa, about 5 minutes. Remove and set aside.

In the same pot, sauté onion, garlic, carrots, and celery until onion is translucent, about 5 minutes.

Add diced potatoes and chicken broth. Bring to a boil, then simmer for 15-20 minutes, or until potatoes are tender.

For a thicker soup, blend part of the soup with an immersion blender, or blend a cup in a regular blender, then return to the pot.

Stir in milk or cream and dried thyme. Season with salt and pepper and simmer for 5 minutes.

Add the browned kielbasa and spinach or kale (if using), and cook until greens are wilted, about 2-3 minutes. Garnish with parsley and serve warm.