In a large pot, heat a little oil over medium heat.
Add the sliced sausage and sauté until browned.
Remove the sausage and set aside.
In the same pot, add onions and garlic; cook until translucent.
Stir in chopped carrots and potatoes; cook for 5 minutes.
Pour in the chicken broth and bring to a boil.
Add the browned sausage back into the pot.
Stir in thyme, salt, and pepper; reduce heat to simmer.
After 15 minutes, add kale or spinach.
Let the soup simmer for an additional 5 minutes before serving.