In a large pot over medium-high heat, combine the tomatillos, garlic, and broth. Bring to a boil. Reduce the heat to medium-low and simmer for 1 minute, or until the tomatillos are cooked through.
Transfer the soup to a blender. Add the oil and puree until smooth. Transfer the soup back to the pot and allow it to cool to room temperature.
Stir in the avocados, cucumber, bell pepper, and onion. Add the cilantro and lime juice. Season to taste with salt and pepper. Refrigerate for at least 2 hours, or until chilled. Keep refrigerated until ready to serve.