Special equipment: deep-fry thermometer
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. Let rest for 15 minutes. Add a little more water if it dries out a bit. (Don't be shy about adding water; the mixture may appear to be too wet to handle at first. If the mixture ends up significantly too wet, add a little more falafel mix.)
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.
Whisk together the tahini, lemon juice and a large pinch of salt in a medium bowl. Whisk in enough water to thin to the desired consistency for a dipping sauce; set aside.
Drop the falafel mixture by heaping tablespoonfuls onto a baking sheet. 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. Enclose the filling with the falafel mixture, and roll the balls until very smooth and there are no signs of cracking. Refrigerate while you heat the oil for frying.
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. Fry the balls in batches, adjusting the heat to maintain the temperature, until golden brown, 3 to 4 minutes. Remove with a slotted spoon to a paper towel-lined plate.
Serve the falafel balls with cucumber slices, lemon wedges and the tahini dipping sauce.
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