01 -
Position a rack in the center of the oven and preheat to 425°F.
02 -
Add the beans, smoked paprika, garlic powder, and onion powder to a large bowl. Stir to combine and set aside.
03 -
Combine the butternut squash, olive oil, a large pinch of salt, and a couple of grinds of black pepper on a small baking sheet. Toss to coat and spread in an even layer. Roast until the squash is tender and deeply browned in spots, about 25 minutes, rotating the baking sheet halfway through.
04 -
Add the roasted squash to the bean mixture and stir to combine. Allow to cool completely.
05 -
Once cooled, use a fork to roughly mash the squash and beans. The mixture should be mostly smooth with small chunks of squash and beans throughout.
06 -
Add the ground beef and 1/2 teaspoon salt to the mixture. Use your hands to mix until combined. Divide into 4 portions and form each into a patty about 4 inches in diameter and slightly more than 1/2 inch thick.
07 -
Heat a large cast-iron skillet over medium-high heat. Once hot, add the canola oil and swirl to coat. Season the patties liberally with salt and pepper on both sides. Add the patties to the skillet and cook until deeply browned and a crust is formed on the first side, 3 to 4 minutes. Flip carefully and cook until the second side is browned, about 2 to 3 minutes.
08 -
Layer 2 half-slices of cheese (1 slice total) on top of each patty. Cover the pan and cook until the cheese is just melted, about 1 minute.
09 -
Layer the patties, lettuce, and tomato on the toasted hamburger buns. Serve with ketchup, Dijon mustard, and mayonnaise on the side.