Season chicken breasts with salt and pepper.
Heat 1 teaspoon olive oil in a large skillet over medium-high heat. Add chicken and reduce heat to medium. Cover and cook until internal temperature reaches 165°F (about 6–8 minutes per side).
Remove chicken, shred, cover with foil to keep warm, and set aside.
In a Dutch oven, heat remaining 2 teaspoons olive oil over medium heat.
Add 1 cup chopped green onions and minced garlic. Sauté for about 2 minutes.
Add diced tomato and cook for 1 minute until softened.
Stir in chicken broth, cumin, and salt. Bring to a boil.
Reduce heat, cover, and simmer for 15–20 minutes.
To serve, layer shredded chicken, diced avocado, remaining green onions, and cilantro into individual bowls.
Ladle hot broth over the layered ingredients.
Serve immediately with a lime wedge on the side.