Preheat oven to 425 degrees.
Heat 1 Tbsp. of oil in a medium pan. Add the onions and cook on medium heat, stirring constantly for 5 minutes until caramelized. Remove from heat and set aside.
Place the mushroom caps, smooth side up, in a shallow dish. In a small bowl, whisk together vinegar, oil, oregano, garlic, salt, and pepper. Pour over the mushrooms. Let stand at room temperature for 15 minutes or so, turning twice. Place caps on a baking sheet and roast at 425 degrees for 20 minutes, until tender.
Heat the same pan you cooked the onions in with a drizzle of oil. Sprinkle the meat liberally with salt and pepper, and gently mix it by hand without breaking up the meat too much. (To test the seasoning, you can fry up a little crumble and taste it.) Form it loosely into patties. Try not to compress the meat too much.
Place the patties on the pan – it should be hot enough that they sizzle. At no point should you squash the burger – keep the juices inside! Cook until the first side is well browned, then flip. Remove the burgers from the pan when they are done to your liking.In a small bowl mash the avocado with 1 Tbsp. olive oil, lemon juice, cumin and salt and pepper to taste. Place mushroom caps on a plate, smooth side down, then top with burgers, caramelized onions, avocado and your favorite burger accompaniments.