Place carrot in a food processor and pulse until finely chopped. Remove and put in a separate bowl.
Add the rest of the ingredients and pulse until combined. Do not over-process— the mixture should be evenly mixed but still rough.
Add in the grated carrot to the mixture and stir until well-combined. Adjust salt to taste.
Heat oil in a frying pan over medium heat. Divide the mixture into patties and fry for 3-5 minutes on each side until golden.
To make tzatziki sauce, finely shred/grate cucumber and squeeze to remove most of the liquid. Place all ingredients in a small bowl and stir until combined. Let sit in the refrigerator for a few hours for the best flavor.
Serve chickpea patties with the sauce, lettuce, red onion, and tomato on burger buns of choice. Patties will store for up to four days in the refrigerator (they are great frozen and reheated too).