With either method, the chickpeas are ready when you can break apart a chickpea with your fingers. Drain before using.
Overnight soak: Place the chickpeas in a medium bowl, add enough water to cover by 1 inch, and let sit uncovered at room temperature for 18 to 24 hours. Quick soak: Place the chickpeas and 2 cups water in a small saucepan. Bring to a boil over medium-high heat and cook for 3 minutes. Remove the pan from the heat, cover, and set aside for 1 hour.
Place the drained chickpeas, scallions, garlic, parsley, olive oil, lemon juice, cumin, coriander, and salt in a large bowl and stir to combine. Transfer to a food processor fitted with the blade attachment. Pulse until the chickpeas are finely chopped, 10 or 12 (1-second) pulses. Scrape down the sides of the bowl with a rubber spatula.
Sprinkle the baking powder over the mixture. Pulse again, scraping down the sides of the bowl as needed, until the mixture forms a ball when you squeeze it in your hand, about 5 to 6 more (1-second) pulses.
Divide the falafel into 6 even portions (about 1/3 cup each), roll each into a ball, and place on a rimmed baking sheet. Add the eggs to the baking sheet.
Line a work surface with an 18-inch-long sheet of plastic wrap, long side facing you. Place 1 falafel ball on the plastic wrap about 3 inches from the left side. Fold the plastic wrap in half from right to left over the falafel ball. Using a rolling pin, roll the falafel into a thin patty about 5 inches in diameter.
Uncover the falafel patty. Place an egg in the center of the falafel patty. Gently pull the plastic wrap up to cover the egg in the falafel patty. Use the plastic wrap to gently smooth out the surface and patch any holes, making sure the egg is completely and tightly encased in falafel. Return to the baking sheet and repeat with the remaining falafel balls and eggs. Place in the refrigerator while the oil is heating.
Place enough oil in a 10- or 12-inch cast iron skillet to come halfway up the sides. Heat over medium-high heat until 350°F on a deep-frying/candy thermometer. Meanwhile, fit a wire rack over a baking sheet; set aside. Stir all the dipping sauce ingredients together in a small bowl and set aside.
When the oil is ready, add 3 of the eggs. Fry, turning every minute or so with tongs, until golden-brown, 4 to 5 minutes total. Remove the eggs to the wire rack. Repeat with the remaining eggs. To serve, cut each egg in half with a serrated knife and serve with the dipping sauce.
