Begin by making the pickled radishes. In a bowl, combine the sliced radishes, white vinegar, sugar, and salt. Stir until the sugar and salt dissolve. Set aside to pickle for at least 15 minutes while preparing the falafel.
Drain the soaked chickpeas and add them to a food processor along with the chopped onion, garlic, parsley, cilantro, cumin, coriander, cayenne pepper, and baking soda. Pulse until you achieve a coarse mixture, scraping down the sides as necessary.
Heat a deep skillet or saucepan with vegetable oil over medium heat. While the oil is heating, shape the falafel mixture into small patties.
Once the oil has reached 350°F (175°C), fry the falafel patties in batches, turning them occasionally, until they are golden brown on all sides, about 4-5 minutes.
Remove the falafel from the oil and drain on paper towels to remove excess oil.
For the tahini drizzle, whisk together the tahini, lemon juice, olive oil, water, and salt in a bowl until smooth. Adjust the consistency with more water if necessary.
To serve, place the falafel on a platter, drizzle with the tahini sauce, and garnish with the pickled radishes.