Position a rack about 4 inches from the broiler heat source, and preheat.
Blend 1/2 cup ice water mixed with ice cubes, 1/4 cup of the sour cream, cucumber, tomato, garlic, scallion whites and 1/2 teaspoon salt in a blender until completely smooth. Remove the small cap on the lid, and slowly drizzle in 1/4 cup of the oil while the blender is running. Season with additional salt if needed. Place the soup in the freezer while you prepare the quesadillas.
Place the black beans and a pinch of pepper in a medium bowl, and roughly mash with the back of a fork. Place 2 of the tortillas on a baking sheet with no overlap. Brush the tops with some of the remaining olive oil, then flip so the oiled side is directly on the baking sheet. Divide the mashed beans between the 2 tortillas, and spread the mixture evenly with a fork or your fingers, leaving a 1/2-inch border around the edge. Sprinkle the cheese evenly on top of the beans. Top with the remaining 2 tortillas, and brush the tops with the remaining oil.
Place the quesadillas under the broiler, and cook until the top tortillas are crisp and lightly golden brown, about 1 minute. Flip the quesadillas with a spatula, and cook until the cheese is completely melted and the tortillas are crisp and lightly golden brown, about 1 more minute. Cut each into 6 wedges.
To serve, ladle (or pour directly from the blender carafe) the soup into 4 chilled bowls. Drizzle with olive oil, add several grinds of pepper and sprinkle with some of the sliced green scallions. Serve 3 quesadilla wedges on the side of each bowl of soup, along with a dollop of the remaining 1/4 cup sour cream.
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