Preheat the oven to 350°F (175°C). Line two baking sheets with foil and lightly spray or oil them.
Spread the drained beans in a single layer on one sheet and the chopped broccoli on the other. Season the broccoli with a pinch of salt and pepper.
Roast both sheets for 12–15 minutes, until the beans begin to split and the broccoli is tender but not browned. Remove and let cool slightly.
While roasting, heat 1½ tablespoons olive oil in a skillet over medium-low. Sauté the diced onion and minced garlic until softened, about 5–6 minutes; set aside to cool.
Pulse the cashews and oats in a food processor until finely chopped; keep the processor bowl for later.
In a large bowl whisk together ¼ cup mayonnaise, the egg yolk, smoked paprika, and ½ teaspoon honey.
Add the sautéed onion mixture, cooled quinoa, roasted beans, and roasted broccoli to the bowl. Add the processed oats and cashews.
Use a potato masher or fork to press and combine the mixture until it starts to hold together.
Transfer about one-third of the mixture to the food processor and pulse a few times until very finely chopped, then return it to the bowl. Stir in the grated cheddar, salt, and pepper and mix thoroughly (hands work well).
Form the mixture into 4–6 patties sized to fit your buns. Refrigerate the patties for about 20 minutes if you want them firmer before cooking (optional).
Heat a large nonstick skillet over medium and add a high-heat oil. Cook the patties 5–6 minutes per side until golden and cooked through, flipping gently. Top with sliced cheddar about 1 minute before removing to melt.
Meanwhile, make the spicy mayo by whisking ½ cup mayonnaise, 2 teaspoons honey, the chipotle pepper, and 1 teaspoon adobo sauce until smooth.
Serve the burgers on buns with spicy mayo and additional sliced cheddar.
Note: These patties are fragile for direct grilling; cook in a skillet or in a cast-iron pan on a grill instead.
