Heat a medium skillet over medium high heat and add oil. When hot, add onion and carrot and cook until vegetables are softened and onions are translucent, about 5-7 minutes. Remove from heat.
Add quinoa, rice, onion and carrots, beets, chickpeas, chickpea flour, chili powder, salt and pepper to a food processor and pulse gently until mixture holds together.
Transfer to a large bowl and using a ½ cup measuring cup, portion mixture out into equal sized mounds, pressing and shaping them into patties with your hands. Depending on the size of your beets and other vegetables, you should end up with 4-5 patties.
Preheat oven to 350 degrees. Coat the bottom of a wide, oven safe skillet with oil and heat over medium heat. Add the burger patties and cook until the first side achieves a browned crust, about 2-3 minutes, then flip and repeat on the second side.
Transfer skillet to oven to finish cooking for about 6-8 minutes, or until patties are warmed through. Remove from oven and make the Harissa spread while the patties cool slightly.
In a small bowl, whisk together vegan mayonnaise, Harissa paste, salt and pepper. Assemble burgers on toasted buns with desired toppings and Harissa spread.