Preheat the oven to 375°F.
Drain and rinse the chickpeas. Dab the chickpeas with a paper towel to remove extra moisture, then place them in the bowl of a food processor. Add the sesame seeds, then process until a paste-like consistency is formed. Scrape the mixture into a bowl.
Add the carrot, red onion, garlic, and cilantro to the bowl of the food processor; pulse until finely ground. Scrape the vegetables into the bowl with the chickpeas.
Stir in the flour, cumin, coriander, kosher salt, and black pepper. Mix with a spoon until combined, then form 4 round patties and place them on a baking sheet. In a medium frying pan, heat ¼ cup oil over medium heat. Carefully add the patties to the pan. If you have one, top with a splatter screen. Fry for about 3 to 4 minutes until golden brown on one side. Flip and fry for another 3 to 4 minutes until golden brown, turning down the heat as necessary.
Transfer the patties to a parchment lined baking sheet. Bake for 20 minutes until golden brown and baked through. Remove the patties from the oven and let them cool for at least 15 minutes to firm up.
To serve, place on a bun with tomato, cucumber, red onion, and lettuce, and top with sauce of choice.