This veggie burger was inspired by the flavors of a samosa. Finely chopped vegetables give the patties texture and the sweet-sour tamarind ketchup lifts up the spices and mashed potato base. If possible, leave the seeds in the chile. The spice is tempered just enough by the bun and mild vegetables, leaving a slight and very pleasant burn.
Make the burgers: Put the potato on a microwave-safe plate and microwave until tender, 7 to 8 minutes. Let cool slightly, then peel. Transfer to a medium bowl; add 1 tablespoon butter, season with salt and pepper and mash with a fork until smooth. Toast the cumin seeds in a large dry nonstick skillet over medium-high heat until golden, 2 to 3 minutes. Pour onto a cutting board and crush with the bottom of a measuring cup.
Heat 1 more tablespoon butter in the same skillet over medium-high heat. Add the cauliflower, carrot, chile, ginger, garlic, curry powder, toasted cumin seeds, 1 teaspoon salt and a few grinds of pepper; cook, stirring, until the cauliflower is dry and the vegetables are crisp-tender, about 3 minutes. Transfer to the bowl with the mashed potato and add the peas; wipe out the skillet and reserve.
Stir the vegetable mixture together to combine. Add the breadcrumbs and knead with your hands until the mixture comes together. Firmly pack into 4 balls, then shape into four 4-inch-wide patties (about 1/2 inch thick). Transfer to a plate and refrigerate until firm, at least 30 minutes.
Meanwhile, make the tamarind ketchup: Stir together the ketchup and tamarind paste in a small bowl and season with salt and pepper.
Cook the burgers: Melt the remaining 2 tablespoons butter in the reserved skillet over medium heat. Add the patties and cook until well browned on the bottom, 3 to 4 minutes. Flip and cook until the other side is browned and the patties are hot in the center, about 3 more minutes. Remove the skillet from the heat and let the burgers stand about 5 minutes while you prepare the buns (this will help them stay together).
Toast the buns and spread with the tamarind ketchup. Serve the burgers on the buns with lettuce, red onion and cilantro.