Slightly trim opposite sides of potatoes; cut each potato lengthwise into fourths, creating four flat ovals or "planks."
In a large pot, bring water and salt to a boil. Add potatoes and return to a boil then reduce heat, cover, and simmer 5 minutes. Using a slotted spoon, remove potatoes to a paper towel-lined platter.
In a large nonstick skillet, heat 1 teaspoon oil over medium heat. Cook half the potatoes 3 to 5 minutes on each side, or until light golden; set aside on serving platter. Repeat with remaining oil and potatoes.
In a small bowl, roughly mash beans with a fork. Add pesto and stir until well blended. Spread equal amounts of pesto mixture on top of each potato slice. Sprinkle evenly with remaining ingredients.