How to Make It
Place potatoes and 1 tablespoon of the salt in a 4-quart saucepan; cover with 6 cups of the water. Bring to a boil over medium-heat, stirring occassionally. Boil potatoes until just tender, about 5 minutes; drain. Cool 30 minutes. Spread potatoes on a parchament paper-lined rimmed baking sheet. Cover and chill until potatoes are cold, about 1 hour.
Meanwhile, cook pancetta in a medium-size nonstick skillet over medium until browned and crisp, about 8 minutes, stirring occasionally. Remove pancetta, and drain on paper towels.
Place egg yolk, garlic, and remaining 2 teaspoons water and 1/2 teaspoon salt in the bowl of a heavy-duty stand mixer fitted with a whisk attachment. Beat on medium-high speed until blended. Gradually add olive oil in a thin, steady stream, beating until aïoli is thickened and smooth, about 4 minutes. Scrape any oil down sides of bowl, and beat 1 minute. Beat in lemon juice until blended. Fold in cheese.
Place potatoes in a medium bowl; gently fold in pancetta, aïoli, and hard-cooked eggs. Top with basil just before serving, if desired.