Preheat the oven to 400 degrees . Line a baking sheet with parchment paper; grease the parchment with the oil.
Using a food processor fitted with the shredding disk, shred the potatoes. In a large bowl, combine the potatoes with enough cold water to cover; let soak for 10 minutes. Drain the potatoes, then layer between kitchen towels. Roll up the towels and squeeze to remove any excess water. Rinse and dry the bowl.
In the bowl, combine the potatoes, onion, eggs, 2 teaspoons salt and 1/2 teaspoon pepper. Arrange the potatoes in 6 mounds on the prepared baking sheet. Using a pastry brush, coat each mound with oil. Bake until golden-brown, about 40 minutes; let cool slightly. Using a spatula, stack 1 mound on top of a second mound; repeat with the remaining mounds to make 3 double mounds. Using your hands, shape each stack into an oval, then press a crevice into the center to create the two lobes of the "brain" (each lobe is 1 serving). Serve with a drizzle of ketchup around each brain.