Combine garlic and lemon juice in a small bowl and let sit 5 minutes.
Whisk yogurt, oil, and sugar in a large bowl; season with salt and pepper. Whisk in garlic mixture.
Using a rolling pin, meat mallet, or the flat side of a chef’s knife, lightly whack cucumbers and radishes so they begin to split apart (but not so forcefully that you smash them to smithereens). Tear cucumbers and radishes into bite-size pieces with your hands and fold into yogurt mixture. Season salad with more salt and lots and lots of pepper.
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