Make the stock: Place chicken in large, heavy stockpot and cover with cold water (at least 16 cup). Bring water to a boil, reduce to a simmer and cook for 30 minutes. Skim off any scum that rises to the surface.
Add onion, carrot, celery, bell peppers, tomatoes, and bay leaf. Continue simmering for 3 hours, adding water if needed to keep chicken covered. Move stock into new bowl.
Make the soup: In a heavy stockpot, heat olive oil over medium heat until hot but not smoking. Add onion and cook until translucent, about 3 minutes. Add garlic and cook 1 to 2 minutes. Add lime juice and stir to reduce, about 1 minute, add 8 cup chicken stock. Bring soup to boil, reduce to a simmer, and cook 15 minutes to reduce slightly.
In a blender, puree soup and return to pot. Heat soup until hot. Taste, adjust seasoning with salt and pepper.
To serve, garnish each serving with a thin slice of lime, 2 radish slices, and chunks of avocado.