1Rinse 1 cup of lentils under cold water and drain. Place lentils in a pot with 3 cups of cold water. Bring to a boil. Then reduce heat and simmer for 20 to 30 minutes until soft (but not mushy). Drain and set aside.
2Combine cilantro, lime juice, lime zest, and 1/4 teaspoon of the salt, and white balsamic vinegar in a small bowl. Whisk in 2 tablespoons of the oil in a slow, steady stream. Set aside.
3Heat 2 teaspoons of the remaining oil in a medium nonstick skillet over medium heat. Add the mushrooms, onion, garlic, and cumin and cook, stirring occasionally, until softened, 4 to 5 minutes. Add the walnuts and cook until the nuts are lightly toasted, 2 to 3 minutes. Transfer to a bowl.
4Combine the ½ cup of cooked lentils and rice in the bowl of a food processor and pulse a few times to break down into a light paste. Add mixture to the bowl of mushrooms and blend. Stir in the parsley, remaining 1/4 teaspoon salt, and the pepper. Form into four 1/2-inch-thick patties.
5Heat the remaining 4 teaspoons of oil in a large nonstick skillet over medium heat. Add the patties and cook, turning once, until browned and heated through, 9 to 10 minutes. Transfer to serving plates and top with any of the optional toppings and the cilantro vinaigrette.