Heat a large heavy-bottomed pan over medium heat. Remove the casings from the sausage and add to the pan along with 2 teaspoons of oil. Cook the sausage until no longer pink, approx. 5-7 minutes, breaking apart the sausage while it cooks.
Using a slotted spoon, transfer the sausage to a plate and set aside. Add the diced onion to the pan and cook for 3-4 minutes until soft and translucent. Add the minced garlic and cook for one minute.
Add the chicken stock and scrape any brown bits that may be on the bottom of the pan. Add the diced tomatoes, tomato paste, thyme, salt, and pepper. Stir to combine.
Bring to a simmer and add the sausage back to the pan and stir in the tortellini. Cook for 2-3 minutes. Add the chopped kale and cook for another 1-2 minutes. The soup is ready when the tortellini floats to the top and is heated through.