To make the burgers: Preheat the oven to 400 degrees. Add all of the burger ingredients except the beans and oil to a food processor and pulse until the ingredients are minced. Add the beans and pulse 3 or 4 times, so it is chunky but not yet a paste. Take a pinch of the mixture and press it together; if it stays together, it's done, and if it crumbles, pulse another 1 or 2 times. If the mixture still doesn't hold together, add 1 tablespoon of water. That should do the trick. (If you don't have a food processor, grate the carrot, onion and garlic into a large bowl. Combine the remaining ingredients in the bowl and mash with a spoon. Repeat the process of testing and pinching some of the mix until it holds together.)
Smear a rimmed baking sheet with the oil. Shape the burger mixture into four equal-size balls, place them on the baking sheet, and gently press them into patties, smooshing any split edges back together. Brush the tops of the burgers with some of the oil from the baking sheet. Transfer the sheet to the oven and bake until the burgers are crisp and browned on the outside, 20 to 25 minutes.
To make the slaw: In a large bowl, whisk together the oil, lime juice, salt and paprika until combined. Add the cabbage and toss with your hands to combine, squeezing the cabbage to start to soften it. Let the slaw sit at room temperature while the burgers bake, or up to 45 minutes.
To serve, divide the burgers among four plates and top with a generous amount of slaw. Serve warm.
“Pantry to Plate,” by Emily Stephenson, uses kitchen staples to cook easy dishes. (Illustration by SEIN KOO/The Minnesota Star Tribune)