In a medium saucepan combine the potatoes and enough water to cover by 2 inches. Bring to a boil. Reduce heat and simmer until potatoes are tender, about 15 minutes; drain potatoes.
When potatoes are cool enough to handle, halve them crosswise.
Place the grated cheese in a shallow dish. Set one potato, cut side down, in the center of the cheese. Press with a spatula to flatten the potato into the cheese. Repeat with the remaining potatoes.
Heat oil in a large nonstick skillet over medium-high heat. Add the potatoes, cheese side down, in a single layer. Don't crowd the pan as they won't crisp up. Cook for 2 to 3 minutes, or until golden. Flip potatoes; cook 3 minutes more. Lightly season with salt and pepper. Repeat with remaining potatoes. Serve warm.
Parker calls grated Parmesan "snow cheese" and she'll ladle a spoonful of it on just about anything. While real Parmigiano-Reggiano can be pricey, a little goes a long way.