Halve, pit, and peel avocados and set aside 1 half. Coarsely chop remaining avocados. In a blender, whirl buttermilk, yogurt, chopped avocados, dill, chopped mint, garlic, and chile until smooth.
Pour soup into a bowl. If you'd like it thinner, stir in about 1/4 cup cold water. Stir in 1 1/2 tbsp. lemon juice and 1 tsp. salt. Chill until very cold, 30 minutes. "It's supposed to be refreshing, like chilled borscht or gazpacho," Fletcher says.
Coarsely shred radishes. Cut reserved half avocado into small dice and sprinkle with a little salt and lemon juice.
Taste soup and add more lemon juice or salt if you like. Ladle into bowls. Place a tuft of radishes and a small spoonful of avocado in center of each serving. Grate a little feta over soup and top with mint leaves. Serve immediately (toppings will sink).