With a chef's knife, cut the snap peas on an extreme diagonal into thirds or halves, discarding any tough stem ends. Place in a plastic bag and refrigerate until ready to serve.
Up to 15 minutes before serving, dress the snap peas: Combine the lemon juice, salt, and sugar in a medium bowl. Spear both garlic halves with a dinner fork. Using this as a whisk, drizzle the oil into the lemon mixture. Add the snap peas and zest and toss to coat. Season with the pepper.
Using a vegetable peeler, shave the Parmigiano into paper-thin strips. Scatter over the snap peas and toss gently before serving.
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