Bring a large pot of unsalted water to a boil. Cook gnocchi according to package directions, in batches if needed, until tender. Reserve 1/2 cup cooking water, then drain well.

Melt 1 tablespoon butter in a wide nonstick skillet over medium heat. Add half the drained gnocchi and cook 3–4 minutes, turning gently, until lightly golden in places. Transfer to a bowl. Repeat with 1 tablespoon butter and remaining gnocchi.

In a separate light-colored small saucepan, melt remaining 2 tablespoons butter over medium heat. Swirl and watch until milk solids turn golden brown and smell nutty, about 2–4 minutes. Remove from heat immediately and stir in ground sage and pepper.

Return all gnocchi to the skillet off the heat. Pour over sage butter and lemon juice. Add Parmesan and 2 tablespoons cooking water; toss gently. Add more water a tablespoon at a time only if needed for a glossy coating. Divide into four servings and serve promptly.

Refrigerate leftovers in shallow containers within 2 hours, or 1 hour above 90°F. Keep at 40°F or below, use within 3–4 days and reheat throughout to 165°F, adding a little water to loosen.