Slice each avocado lengthwise around the pit, twist to separate the halves, and remove the pit. Scoop the flesh out with a large spoon and place it directly into the blender jar, discarding any dark or fibrous spots.
Roll each lime firmly on the counter for 10 seconds, then cut in half and juice using a citrus juicer into a small bowl, yielding 3 tablespoons total. Set aside.
Roughly chop the garlic clove. If using jalapeño, slice it in half, scrape out the seeds and membrane, and roughly chop. Measure out the cilantro, cumin, salt, and white pepper and have them ready.
Add the cold vegetable broth, lime juice, garlic, jalapeño (if using), cilantro, cumin, salt, white pepper, and olive oil to the blender with the avocado. Blend on high for 60 to 90 seconds until completely smooth, scraping down the sides as needed.
With the blender running on low, add the cold water gradually — starting with ¼ cup — until you reach your preferred thickness. Taste and adjust lime juice, salt, or cumin as needed, blending briefly after each addition.
For an ultra-silky texture, pour the blended soup through a fine mesh strainer, pressing it through with a spatula. Skip this step if a slightly more rustic texture is fine.
Transfer the soup to an airtight container and press plastic wrap directly onto the surface before sealing. Refrigerate for at least 30 minutes, or up to 1 hour for best flavor and color.
Stir the chilled soup and ladle into cold serving bowls. Top with diced tomato, sliced radish, fresh cilantro, a drizzle of olive oil or crema, crushed tortilla chips, and a pinch of flaky sea salt. Serve immediately.