Peel the carrots and remove the leaves. Chop them into quarters and add to a food processor. Pulse until finely minced. Place the minced carrot into a large bowl.
Trim the eggplant stems and cut into quarters so they fit in the food processor. Pulse until finely minced, then transfer to the bowl with the minced carrots and mix well. Season with salt and pepper.
In a large skillet over high heat, add the olive oil. Add the vegetable mixture to the skillet and sauté until most of the moisture has cooked off, about 15-20 minutes. Stir constantly. When no liquid spreads on the pan between stirs, then it is ready. It should appear slightly dry and browned, similar to ground beef. Wipe the skillet clean.
Transfer the cooked mixture to a bowl and cool until it is no longer steaming. Stir in the spring onions, eggs, shredded goat cheese and flour. Mix well to combine.
Wet your hands, then use a 1/2 cup measuring cup to measure out evenly sized balls of the mixture. Form into patties and place on a plate lined with parchment or a silicone mat.
Lightly grease the skillet with more olive oil. Add the patties and sear on each side for 3-4 minutes or until completely cooked through. The patties should crisp up on the edges and look similar to a regular burger.
Assemble the burgers on sesame buns with tomato, mayonnaise, lettuce or kale. Serve and enjoy!