In a large saucepan, add the potatoes and enough water to cover by 1 inch. Bring to a boil, lower the heat and simmer until the potatoes are tender, about 20 minutes; drain, reserving 1/3 cup of the potato cooking water. Return the potatoes to the pot and mash well. Add the cooking water and 3 tablespoons olive oil; season with salt and pepper.
Position a rack in the center of the oven and preheat to 425 degrees . Line a baking sheet with parchment. In a large nonstick skillet, heat 1 1/2 tablespoons olive oil over medium heat. Add the beet, carrots and scallions to the pan and cook, stirring occasionally, until softened, about 5 minutes. Transfer to a large bowl and add the rice, lentils, rice cereal, sunflower seeds and ketchup. Mash with your hands until it resembles a meat loaf mixture. Season with salt and pepper, then shape into six 3 1/2-inch patties.
Using the same skillet, heat 2 tablespoons olive oil over medium heat. Add half of the patties and cook, turning once, until browned, 6 to 8 minutes. Transfer to the prepared baking sheet; repeat with the remaining patties.
In a small bowl, toss the peas with salt, pepper and 1 1/2 teaspoons olive oil. Divide the mashed potatoes into 6 portions. Shape each portion into a mound, form a well in the center and place on top of the patties. Fill each well with some of the peas. Bake until hot, about 20 minutes. Drizzle with the remaining 1 tablespoon olive oil.