Preheat the oven to 375°F and line a baking sheet with parchment paper.
Place the mushrooms in a food processor and pulse 5-7 times until finely chopped. Set aside. If you don’t have a food processor, you can finely chop by hand.
Sauté the onion in a medium-sized non-stick pan with a few tablespoons of water until the onions are soft, about 5-7 minutes. Add water as needed to prevent burning. Add the mushrooms to the pan and continue to cook on medium heat until the mushrooms soften (another 5 minutes) and begin releasing their juices.
Drain any additional liquid from the pan (it’s important to not add additional liquid to the processor) and transfer the onion/mushroom mix to the food processor. A fine mesh strainer works well to drain off any extra liquid.
Add all the other ingredients (reserving ½ cup oats) to the processor and process until just combined but not smooth – you want to maintain some texture. Stop and scrape down the sides of the processor container as needed. If you don't have a food processor, mash all ingredients together with a potato masher or fork.
Transfer the mix to a large bowl and add the remaining ½ cup oats and mix.
Scoop out a little less than ½ cup and shape into a burger and place on the baking sheet. Repeat with remaining mix. Bake the burgers for 30 minutes. Flip and bake for another 10 minutes. Remove from the oven and let sit covered for 10 minutes. See notes if making balls instead of burgers. Create your burger the way you like it! See above for my preferred toppings.