Pulse carrots in a food processor until finely chopped, but not pureed.
Heat 1 tablespoon of olive oil in a non-stick skillet. Add carrots and 1/4 teaspoon kosher salt. Cook carrots on medium-low until softened (approximately 5-7 minutes), stirring occasionally. Allow to cool 1-2 minutes. While carrot mixture is cooking, rinse out the food processor and set aside.
After the carrots have cooled for 1-2 minutes, add them to the food processor along with chickpeas. Process on low until well combined but not pureed, scraping down the sides as needed.
Transfer chickpea mixture to a large bowl. Add vegan mayonnaise, chickpea flour, nutritional yeast, parsley, garlic powder, onion powder, dill weed, 1/2 teaspoon kosher salt and white pepper. Stir to combine.
Use your hands to form four patties. Heat 2 tablespoons of olive oil in a non-stick skillet. Add burgers to the skillet and cook over medium heat for 5-7 minutes. Carefully flip burgers and cook for 5-7 more minutes.
Serve with desired toppings.