Swirl enough oil in a medium skillet to coat the bottom of the pan. Heat over medium-low flame until shimmering. Add the broccoli, carrots, and onion. Season with salt and pepper. Saute until tender, about 5 minutes.
Add the eggs to a deep bowl, and beat them lightly. Add the sautéed vegetables, quinoa, breadcrumbs, cheese, cumin, parsley, salt & pepper. Stir until well combined.
Add another swirl of oil to the skillet, and heat over medium-low flame until shimmering again. Scoop a small fistful of the quinoa mixture with your hand, squeezing it slightly to make it compact (an ice cream scoop works really well here, and dipping your hands in water so the quinoa doesn’t stick is helpful, too). Gently flatten the ball into a 2-inch patty. Place in the skillet. Repeat with the remaining mixture. Cook until golden and crisp underneath, 3 to 4 minutes, then flip, and cook until golden and crisp again, 3 to 4 minutes more. Transfer to a paper-towel lined dish to drain for a few minutes before serving. May be served hot, warm, or even cold.