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Spread several layers of paper towels on a clean work surface. Spread grated cucumber on the paper towels, and sprinkle with 1/2 teaspoon salt. Allow to stand about 10 minutes.
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Roll up cucumber inside the paper towels, like a jelly roll. Squeeze tightly to remove as much liquid as possible. Discard liquid.
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Place cucumber shreds in a large mixing bowl, and add garlic, red wine vinegar, and olive oil. Stir, blending well.
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Add Greek yogurt, black pepper, remaining 1/2 teaspoon salt, and fresh dill. Stir until ingredients are well blended and refrigerate until ready to serve.
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To assemble the bowls, place 1 1/2 cups thinly sliced Romaine in each serving bowl. Add 1/2 cup cooked chicken, 6 cherry tomatoes, 1/4 cup kalamata olives, 1/4 chopped bell pepper, and 1 thin slice red onion to each bowl.
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Top with each bowl with 1 tablespoon crumbled feta, or more, if desired. Add 1 peperoncini for garnish.
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Stir tzatziki, taste, and adjust seasoning, if needed. Add enough olive oil to create a pourable sauce; pour over each bowl. Serve each bowl with 2 or 3 tablespoons tzatziki, or to taste. Store any unused tzatziki in a sealed container in the refrigerator, 2 to 3 days.