Make the burgers: Drain and rinse the chickpeas. Blot them with a paper towel or clean dish towel to remove any extra moisture, then place them in the bowl of a large food processor. Add the sesame seeds and process for a minute or so until a paste-like consistency is formed (if necessary, stop and scrape the sides of the bowl, and process again). Scrape the mixture into a large bowl.
Peel the garlic, red onion, and carrots. Chop the onion and carrot in rough chunks. Add the garlic to the food processor and process until finely chopped. Add the carrot and onion and pulse several times until finely chopped. Add the cilantro to the processor and pulse a few more times until chopped.
Scrape the vegetables into the bowl with the chickpeas. Stir in the flour, cumin, coriander, cayenne, kosher salt, and black pepper. Mix with a spoon or with your hands until fully combined, then form 8 round patties and place them on a baking sheet.
In a large skillet, heat 1 tablespoon olive oil over medium-high heat. Add 4 of the burgers and fry them for 1 to 2 minutes per side, until golden brown. Remove the cooked burgers and place them back on the baking sheet. Wipe out the pan, add an additional drizzle of oil, and fry the remaining 4 burgers, keeping in mind that the pan will be hotter and the cooking time slightly quicker for the second batch.
Once all burgers are browned, place them in the oven on the baking sheet and bake 15 minutes on one side, then flip and bake another 15 minutes. Remove from the oven and place on a wire rack to cool for at least 10 minutes.
Make the sauce: Peel and finely mince the garlic. Chop the dill. Juice the lemon. In a small bowl, stir together the garlic, dill, lemon juice, Greek yogurt, olive oil, kosher salt, and 2 tablespoons water. Store leftovers in a sealed container in the refrigerator for to 2 weeks.
Assemble the burgers with the yogurt sauce and desired fixings.
Store leftover burgers refrigerated for up to 5 days, or frozen in an airtight container or plastic bag. To reheat, heat straight from the refrigerator or freezer in a 375°F oven until crispy and heated through; about 8 minutes for refrigerated burgers and about 15 minutes for frozen burgers.
Recipe by Love and Lemons at https://www.loveandlemons.com/falafel-burger-recipe/