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Special equipment: deep-fry thermometer
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Put the falafel mix in a large bowl, and add 1/4 cup of water at a time, mixing well with your hands after each addition, until the mixture is wet and tacky and malleable enough to be squeezed and rolled without cracking.
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Let rest for 15 minutes.
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Add a little more water if it dries out a bit.
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(Don't be shy about adding water; the mixture may appear to be too wet to handle at first.
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If the mixture ends up significantly too wet, add a little more falafel mix.)
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Meanwhile, mix the feta, roasted peppers, parsley, Parmesan, olive oil, vinegar and mint with a spoon in a small bowl to combine, breaking up the bits of feta; set aside.
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Whisk together the tahini, lemon juice and a large pinch of salt in a medium bowl.
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Whisk in enough water to thin to the desired consistency for a dipping sauce; set aside.
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Drop the falafel mixture by heaping tablespoonfuls onto a baking sheet.
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Roll each into a ball, hold in the palm of your hand, make a deep indentation with your finger and fill it with about 1/2 teaspoon of the feta-pepper filling.
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Enclose the filling with the falafel mixture, and roll the balls until very smooth and there are no signs of cracking.
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Refrigerate while you heat the oil for frying.
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Fill a large pot with about 2 inches of vegetable oil, attach a deep-fry thermometer and bring the oil to 375 degrees F over medium heat.
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Fry the balls in batches, adjusting the heat to maintain the temperature, until golden brown, 3 to 4 minutes.
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Remove with a slotted spoon to a paper towel-lined plate.
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Serve the falafel balls with cucumber slices, lemon wedges and the tahini dipping sauce.