Bring water, lentils, and salt to a boil in a medium saucepan over high heat. Reduce the heat to medium-low and simmer, stirring occasionally, until the lentils just begin to fall apart, about 25 minutes. Drain the lentils well, transfer to a paper towel–lined baking sheet, and gently pat dry to remove excess moisture.

Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Add the mushrooms and cook until the liquid they release has evaporated and they are softened, about 8 minutes. Stir in the onions, celery, leek, and garlic and cook until the vegetables are softened and just beginning to brown, 10 to 15 minutes. Transfer this mixture to the baking sheet with the lentils and let cool to room temperature, about 30 minutes.

Combine water, bulgur, and salt in a large bowl. Cover and microwave until the bulgur is softened, about 5 minutes. Drain thoroughly in a fine-mesh strainer, using a rubber spatula to press out every bit of excess moisture, then return to the bowl to cool slightly. Pulse the cashews in a food processor until they are finely ground, about 25 pulses.

Stir the ground cashews, the cooled lentil-vegetable mixture, and mayonnaise into the bowl with the bulgur until combined. Transfer half of the mixture back to the food processor and pulse until coarsely ground, about 15 pulses—the mixture should be cohesive but still look roughly textured. Combine the processed half with the remaining mixture in a separate bowl and stir in the panko, salt, and pepper. Divide into 12 equal portions and pack tightly into patties about 1/2-inch thick.

Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Add 4 patties to the skillet and cook until well browned on the first side, about 4 minutes. Flip the burgers, adding a splash more oil if the skillet looks dry, and cook until the second side is deeply browned, about 4 minutes. Transfer to a plate and tent with foil while repeating with the remaining patties. Serve hot on buns.