Heat a large pot over medium-high heat. Add the oil and sear the chicken breasts for 3 to 4 minutes per side until golden. Remove and set aside.
Add a drizzle of oil if needed. Add the diced onion, garlic, and 2 diced green chiles. Cook stirring often until softened and translucent.
Turn the oven to broil. Place the 3 remaining green chiles on a sheet pan, spray with oil, and broil until charred and blistered on one side. Flip and repeat. Remove from the oven, let cool, peel off the blistered skin, chop, and add to the pot.
Pour in the bone broth and return the seared chicken to the pot. Cover and boil for 10 to 12 minutes until the chicken is fully cooked. Remove the chicken and shred.
While the chicken is out of the pot stir the softened cream cheese into the soup until fully melted and incorporated.
Add the salsa verde, corn, lime juice, all seasonings, shredded chicken, and tortellini. Bring to a boil and cook until the tortellini are fully cooked through.
Serve immediately topped with sour cream, avocado, and jalapeño slices.