Chop the carrots, shallot and parsley in a food processor on the pulse setting until finely chopped, but not pureed. You may have to stop to scrape down the sides.
Heat 1 tablespoon of olive oil in an 8-inch non-stick skillet. Add carrot mixture and salt.
Cook carrot mixture on medium-low for 10 minutes, stirring occasionally. Allow to cool 1-2 minutes.
While carrot mixture is cooking, rinse out the food processor and set aside.
Add chickpeas and carrot mixture to the food processor. Process on low until everything is well-combined but not pureed, scraping down the sides as needed.
Transfer chickpea mixture to a large bowl. Add 1 tablespoon chickpea flour and 1 tablespoon Buffalo sauce. Mix thoroughly using a spoon or your hands. If you find the mix is too loose, you can mix in another tablespoon of chickpea flour.
Use your hands to form 4 burger patties.
Heat 2 tablespoons of oil in a 10-inch non-stick skillet.
Add burgers to the skillet and cook over medium to medium-low heat for 5 minutes.
Using a turner spatula, carefully flip burgers and cook for 5 more minutes.
Add ½ teaspoon Buffalo sauce to the top of each burger and carefully flip. Cook over low heat for 1 minute.
Add an additional ½ teaspoon Buffalo sauce to the top of each burger and carefully flip. Cook an additional minute over low heat.
Serve with desired toppings.