Preheat the oven to 350ºF. Line a large baking sheet with parchment paper and set aside.
Add the olive oil to a large skillet set over medium heat. When hot, add in the mushrooms and cook until tender and the liquid released has been cooked off, about 8 to 10 minutes. Remove from the heat and set aside to cool slightly.
Meanwhile, add the beans to the bowl of a food processor. Pulse until about half of the beans are pureed, but the rest still have some texture. Add to a large bowl, along with the onion, garlic, parsley, bread crumbs, flour, oregano, salt and pepper. Add the mushrooms to the bowl of the food processor (save the skillet for later) and pulse until finely chopped. Add to the large bowl and mix to combine. Taste and season with additional salt and pepper if needed.
In a small bowl, beat together the egg, tomato paste and worcestershire with a fork. Stir into the bean / mushroom mixture. Pat the mixture into four large patties (about 4 inches in diameter) on the prepared baking sheet.
Bake for about 25 to 30 minutes, until firm and lightly browned.
To make the onions:
Meanwhile, carefully wipe out the large skillet you used for the mushrooms. Add the olive oil and set it back over medium heat. When hot, add in the onion, salt and pepper. Cook, stirring often, until lightly golden and slightly caramelized, about 15 minutes, turning down the heat to medium-low / low as needed so that they do not burn.
To make the sandwiches:
Heat a large griddle or a large nonstick skillet over medium heat.
Set four slices of the bread on a work surface. Top each with a burger patty, a slice of cheese, some of the onions and finally another piece of bread. Spread some of the butter on the top and bottom of each sandwich.
Transfer the assembled sandwiches to the heated griddle and cook, carefully flipping once halfway through, until golden brown on each side and the cheese is melted.