Make flax egg: Mix ground flaxseed and water. Set aside.
Place pan on medium heat. Once hot, add olive oil and allow to evenly coat the pan.
Add onions and cook until translucent, mixing frequently.
Add broccoli and cook for 3-5 minutes. Season with salt. Add garlic and saute for 30 seconds. Remove from heat.
In a food processor, blend oats until you have a coarse flour consistency. Add carrots, chickpeas, nutritional yeast, parsley, flax eggs, and broccoli mixture. Pulse until desired consistency is reached. Add salt and pepper and adjust seasonings as needed. If the mixture is too wet, add more dry ingredients like ground oats or nutritional yeast. If the mixture is too dry, add more wet ingredients like an additional flax egg.
Place mixture in the fridge to chill for 30 minutes (optional).
Using the same pan from the broccoli, place on stove on medium heat.
Spray the pan with non-stick cooking spray or a small amount of oil and cook patties for 3-4 minutes on each side. They should be a nice golden-brown color.
Serve on hamburger buns with lettuce, tomato, red onion, and vegan mayonnaise.
To bake: Preheat the oven to 375 degrees Fahrenheit. Add patties to a parchment paper-lined baking sheet and bake for 20 minutes.