In a large nonstick skillet over medium heat, warm 1 tablespoon olive oil.
Add onion to a blender or food processor and blend until finely chopped. Transfer to the pan. Add the mushrooms to the blender and blend until finely chopped. Transfer to the pan along with ½ teaspoon salt. Cook until all of the water has evaporated, about 7-9 minutes, stirring occasionally.
Add the garlic, smoked paprika, beans, and remaining ½ teaspoon salt. Cook for about 1 more minute, stirring often to make sure not to burn the spices or garlic.
Add oats to the blender and blend into a flour. Add half of the bean mixture to the blender (about 1 cup) along with the rice and egg. Blend until a thick paste forms (don’t over blend). Scrape under the blade if needed. Transfer to a bowl and stir in the remaining bean mixture from the pan. Spread the mixture around the bowl to help it cool faster, place in the freezer for about 10 minutes to cool off.
Form into four ½-inch thick patties. Wipe out the skillet. Warm remaining 2 tablespoons oil in the skillet over medium-high heat. Once the oil is hot and shimmering, add the patties to the pan. Press each patty down gently with a spatula and cook for 5 minutes, or until browned and crispy. Flip, press, and cook for 3 more minutes on the other side, or until browned.
To make the special sauce, add all ingredients to a small bowl and mix to combine.Assemble your burgers and devour!