Add olive oil (or high-heat oil of your choice) to a skillet over medium heat. Once the oil is warm, add the chopped onions, and sauté until soft, about 1 minute. Add the chopped mushrooms and a pinch of salt, and cook until they turn brown, 6 to 10 minutes, turn down the heat a little, as needed.
Add the vinegar, maple syrup, and tamari (gluten-free soy sauce). Stir, lower the heat, add the smoked paprika, hot sauce(or sriracha), garlic, and wore. Remove the skillet from the heat and let cool, about 1 minute.
To a food processor, add the brown rice, flaxseed, 1/2 cup of bread crumbs, and sautéed mushroom mixture. Pulse until ingredients are combined. The veggie burger mixture should hold together in your hand but still have some texture.
Transfer the mixture to a large bowl and add the remaining breadcrumbs.
Form into patties, place them in a container and allow them to form in the fridge for 1 hour.
If you're grilling the burgers, preheat your grill to medium-high heat. Brush the patties with avocado oil and spray the grill with cooking oil spray. Place the burgers on the grill and use a spatula to press down lightly. Grill for 6 to 7 minutes on the first side, flip, and grill for 6 to 7 minutes on the second side, or until charred and cooked through. Optional: Grill buns to your likeness, if desired.
Instructions for cooking the burgers on the stovetop. Heat skillet over medium heat. Coat the bottom of the skillet with olive oil and cook the burgers for 5 to 6 minutes per side, or until charred and cooked through.
Remove from the heat and serve with desired toppings.