In a large Dutch oven, heat the olive oil over medium-low heat. Add the onion and cook, stirring occasionally, until softened, about 7 minutes. Add the garlic; cook for 3 minutes. Add the chicken thighs and broth, bring to a boil and simmer for 4 minutes. Stir in 1 cup salsa, cover and remove from the heat. Let stand until the chicken is cooked through, about 45 minutes. Transfer the chicken to a cutting board and shred.
Return the broth to a simmer; season with salt. Divide the chicken among 4 soup bowls and top with the avocado slices. Ladle the broth into the bowls and top each portion with the cilantro, sour cream and remaining salsa.