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Combine the carrots and lentils in a saucepan, cover with 21⁄2 cups of lightly salted water, and bring to a boil. Reduce the heat and cook for 15 minutes, uncovered, or until the lentils start to fall apart. Drain through a fine sieve, pressing out excess moisture with the back of a spoon. Spread in a large bowl and set aside to cool.
Heat 2 tablespoons of oil in a skillet over medium heat, add the onion, and cook for 5 minutes, or until soft. Add the garlic and curry powder and cook for 1 minute more. Remove from the heat.
Coarsely chop the cashews in a food processor.
Add the lentil mixture, onion, and egg yolk, and pulse the mixture to combine, leaving a bit of texture.
Transfer to a bowl and add the bread crumbs, salt, and pepper, to taste. Shape the mixture into six to eight patties of a 3⁄4- to 1-inch thickness. Place them on a plate lined with plastic wrap. Refrigerate for 1 to 2 hours.
Heat 1 tablespoon of oil in a nonstick skillet over medium heat, add the patties, and cook for 3 minutes per side, or until browned.
Combine the mayonnaise, curry powder, and lemon juice in a small bowl and mix to blend. Adjust the curry to taste. Serve the patties with or without buns, with the sauce.