In a small saucepan, combine the quinoa with ½ cup water. Bring to a boil, add ¼ tsp salt, then reduce the heat to low and cover the pan. Cook gently for 18 to 20 minutes, until the water is absorbed, the grains are tender and the gem of the quinoa is exposed. Set aside uncovered to cool as you prepare the remaining ingredients.
Scrub the beets and carrots. Trim off the ends and any hairy bits from the beets, but otherwise it's not necessary to peel them. Grate both vegetables using the large holes in a box grater.
In a medium skillet, warm the olive oil over medium heat. When hot, add the onion, cumin and remaining ¾ tsp salt, and cook until the onions are softened and beginning to caramelize, about 7 to 10 minutes. Add the garlic, stirring until fragrant, and then the beers and carrots. Cook, stirring periodically, until the vegetables are tender and have concentrated and collapsed a bit, 6 to 10 minutes. Some caramelization and blistering on the carrots and beets is good! Deglaze the pan with the vinegar, using a wooden spoon to scrape any browned bits, then scrape the mixture into a mixing bowl and set aside to cool slightly.
Stir the quinoa into the vegetables, along with the potato flakes, starch and hazelnuts. Shape into 6 medium-sized burgers.
Place a wide skillet over medium heat, and once warm add a splash of oil. Add as many burgers as will fit comfortably without crowding the pan (usually 3 burgers will fit into a 10-inch/25 cm skillet) and cook until browned and crisped on the bottom, 5 to 7 minutes, then flip and repeat on the other side. The burgers will firm up a bit as they cook, and then further once they're removed from the heat and have cooled slightly. Serve warm.