Step 1

Position a rack in the center of the oven and preheat to 425 degrees F.

Step 2

Add the beans, smoked paprika, garlic powder and onion powder to a large bowl. Stir to combine, then set aside.

Step 3

Combine the butternut squash, olive oil, a large pinch salt and a couple grinds black pepper on a small baking sheet. Toss to coat and spread in an even layer. Roast in the oven until the squash is tender and deeply browned in spots, about 25 minutes, rotating the baking sheet once halfway through the cooking time.

Step 4

Add the squash to the bean mixture and stir to combine. Allow to cool completely.

Step 5

Once cooled, use a fork to roughly mash the squash and beans. It should be mostly smooth but with small chunks of squash and beans throughout. Add the beef and 1/2 teaspoon salt and use your hands to mix until combined. Divide into 4 portions and form each into a patty about 4 inches in diameter and a little more than 1/2 inch thick.

Step 6

Heat a large cast-iron skillet over medium-high heat. Once the pan is hot, add the canola oil and swirl to coat. Season the patties liberally on both sides with salt and pepper and add to the pan. 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 also browned, 2 to 3 minutes. Layer 2 pieces of the cheese (1 slice total) on top of each patty, then cover the pan and cook until cheese is just melted, about 1 minute.

Step 7

Layer the patties, lettuce and tomato on the toasted hamburger buns and serve ketchup, mustard and mayonnaise alongside.