Preheat the oven to 375 and line a baking sheet with foil. Spray with cooking spray.
In a large non-stick skillet over medium heat saute the onion in the olive oil until starting to get tender. Add the broccoli and carrots and contine to saute until both are starting to get tender. Sprinkle the mixture with salt and pepper to taste.
Mix the vegetable mixture, cooked quinoa, eggs and cheddar cheese in a bowl. Add salt about a ½ teaspoon of salt. Mix well.
Using a ¼ meansuring cup carefully make the patties and place them on the baking sheet.
Bake the patties for 10 minutes. Pull them out of the oven and flip them over. Bake for another 8-10 minutes.
Serve these as a vegetarian main course, as a topping to a salad or even the base for a fabulous veggie wrap.