Roast the peppers. You can do this on a sheet tray under the broiler or in an air fryer. This should take around 10 minutes, depending on the method you're using. Turn peppers as needed so all sides are charred. As soon as they're done, place them in a large bowl and cover with plastic wrap. Let sit 10-15 minutes to let them steam. Peel the skins off, cut open and remove the seeds and stem.
While you are cooking the peppers and letting them cool, make the potato mixture. In a small saucepan, add the potatoes, salt and enough water to just cover them. Heat over medium and bring to a boil, reduce heat to medium/low, cover and cook 10 minutes or until fork tender. Drain and add the almond milk. Start with 1 1/2 cups and add more if needed. Use an immersion blender and blend until smooth and creamy. Set aside until needed.
Add the cooked Poblano peppers, green onion, garlic oil, and salt to a large stock pot. Cook for 3 minutes on medium heat. Add in the broth (or water) and turn the heat off. Using an immersion blender, blend until smooth. Add the raw chicken and cover and cook until chicken is cooked through- about 13 minutes.
Remove the chicken to a cutting board. Add the potato mixture to the soup mixture and stir. Chop up the chicken and add back to the soup. Stir and serve warm.