Add avocado oil to a medium nonstick skillet over medium heat. Once hot, add the onion. Season with 1 1/2 tsp kosher salt & sauté, stirring occasionally, 5-7 minutes, until onion is softened. Reduce heat to low. Add black beans to the skillet, stirring to combine. Cook, stirring occasionally, for 8-10 minutes, until the black beans are toasted. Add garlic, chili powder, & smoked paprika to the skillet. Stir to coat the onion & black bean mixture, cooking for 1-2 minutes, until fragrant. Remove the skillet from the heat & set aside.
Meanwhile, as the black bean mixture cooks, prep the oats & walnuts. Add the oats to a food processor. Process until the oats are broken down into a fine, flour-like consistency. Add the walnuts to the food processor. Pulse to chop into small pieces. Set aside.
Add the egg, mayonnaise, tomato paste, & Worcestershire sauce to a small bowl. Whisk to combine. Set aside.
Add the cooked quinoa, black bean mixture, oat & walnut mixture, & wet ingredients to a large mixing bowl. Use a spatula to fold, combining the mixture evenly. Take care not to overwork the mixture, crushing the black beans & quinoa, as you fold it together.
Divide the mixture into 4 even portions & shape into 1-inch thick burger patties. I find it’s easiest to roll each portion into a compact ball first, then pressing it into a patty shape. It’s important that your burger patties are as compact as possible; this will help prevent them from crumbling or falling apart on the grill. Transfer to the prepared baking sheet. Transfer the black bean burger patties to the freezer to chill for 20-30 minutes.
Preheat the grill. Immediately prior to grilling, grease the grill grates. Generously oil a paper towel (or spritz it with nonstick cooking spray). Use the grill tongs to run the oiled paper towel over the grill grates. Remove the black bean burgers from the freezer & place on the prepared grill grates, directly over the fire. Grill for 4-5 minutes per side, flipping once halfway through. If desired, place a slice of cheese on each burger with 2 minutes cooking time remaining.