Preheat oven to 425 degrees F. Make sure all of your ingredients are prepped, things will move quickly once cooking begins. Line a baking sheet with parchment paper, optional for easy clean up.
Toss the cauliflower in 2 tablespoon of the olive oil and season with salt and pepper. Spread out on the baking sheet and bake for 15-20 minutes or until browned and beginning to char. Let cool slightly.
Meanwhile, combine the quinoa and the veggie stock in a small saucepan with fitted lid. Bring to a boil. Reduce heat to low, cover, and simmer 15 minutes, or until liquid is absorbed. Let cool slightly.
Meanwhile, melt the butter in a cast iron skillet over medium heat. Add the onion and saute until starting to brown, about 4-5 minutes. Add the mushrooms and garlic, and saute until soft, about 6-8 minutes. Remove from skillet, set aside.
In a large bowl, combine the quinoa with the cauliflower. Add the parsley, thyme, and panko and stir or mix with your hands. Season very generously with salt and pepper to taste. Add the egg and mix until evenly combined. Form 5 round balls with your hands. Make sure to squeeze any moisture out of the balls and make sure that they are intact and holding together. If not, add a bit more panko. Carefully form patties out of the balls, keeping the mixture squeezed together.
Heat the remaining olive oil in the same skillet over medium high heat. Add the burgers and cook until fully browned on the bottom, about 5 minutes. Carefully flip the burgers. Top the burgers with the cheese, cover, and cook until browned and cheese is melted, about 5-8 minutes.
To serve, place burger patties on the bottom burger buns. Evenly distribute mushroom mixture. Top with fresh arugula, mayonnaise or other toppings of your choice. Top with burger bun tops. Enjoy!