Brown chicken: Season the chicken with the salt and pepper. Heat the oil in a Dutch oven or large soup pot over medium-high heat until shimmering. Place the chicken in the pot, skin-side down, and cook until browned, 4 to 5 minutes. Flip the chicken and cook 4 minutes more.
Add broth and vegetables: Add the broth, onion, carrots, celery, thyme sprigs, and bay leaves. Stir to combine using a wooden spoon and scrape any browned bits from bottom of the pan.
Bring to boil: Bring the broth to a boil, cover partially, and reduce the heat to low. Simmer until the chicken is cooked through (up to 30-60 minutes for legs, or until chicken reaches internal temperature of 165°F (74°C). The cooking time will vary depending on the size and thickness of the drumsticks).
Shred chicken: Discard the thyme sprigs and bay leaf. Remove the chicken from the soup and transfer to a cutting board. Cut the chicken off the bone, using two forks to shred it into bite-size pieces.
Add back to soup: Return the chicken to the soup and bring it to a simmer to heat it up while you cook the tortellini.
Cook tortellini: Cook up the tortellini (check the package instructions for timing, and cook a minute less). I prefer to cook it in a separate pot. Cook until it is not quite al dente and drain. The tortellini will continue cooking in the hot broth so be sure not to overcook them. IF you plan to eat the whole pot of soup, you could cook the tortellini in the broth. IF not, the tortellini will get mushy if you store it in leftover broth.
Serve up the soup: Add cooked tortellini to bowls while and ladle in the soup. Taste and if needed, add any seasonings if desired, including up to ½ teaspoon of salt.
Add more herbs: add any extra chopped parsley or dill.