Trim the ends of the both cucumbers off and discard the ends, then cut the cucumber into large chunks and add to the blender jar (yes, skin and any seeds and all).
Add the water, Marcona almonds, garlic, vinegar, and salt to the blender jar as well, then blend, ramping up to high speed, until smooth, about 1-2 minutes.
Turn the speed of the blender down to low, then slowly drizzle in the ⅔ cup olive oil and the 2 tablespoons lemon olive oil.
Use a fine-mesh strainer to strain the soup, using a ladle to force the liquid through. (Discard any remaining solids from the strainer.) Chill the soup until cold in the refrigerator, at least 1 hour.
Just prior to serving, carefully shake the soup in its container or reblend it (it will have separated slighly, more so the longer you've left it; this is completely natural, expected, and nothing to fret over). Taste the soup for seasoning at this point; depending on how salty your almonds were, you may need to add a little more salt, a little lemon, or a little vinegar.
Serve the chilled soup, garnished with a little spoonful of diced peaches, a pinch of Aleppo or Esplette pepper, and a small drizzle of lemon olive oil.