Preheat oven to 350°F. Place sweet potatoes directly on oven rack in preheated oven, and roast, turning occasionally, until softened, 1 hour to 1 hour and 30 minutes. (Place a baking sheet on bottom rack to catch drips, if desired.) Cut potatoes in half lengthwise; let cool 1 hour.
Scoop 1 pound sweet potato flesh (about 2 1/4 cups) into a large bowl. Reserve remaining sweet potatoes for another use. Discard skins. Cover and chill flesh until cold, at least 2 hours or up to 2 days.
Stir together all-purpose flour and whole-wheat flour in a small bowl. Using a fork, mash chilled sweet potato flesh in a separate bowl until mostly smooth; stir in egg, 1 ounce Parmesan (about 2/3 cup), and 1 teaspoon salt with a fork until well combined. Stir in flour mixture until just incorporated
Turn dough out onto a well-floured work surface; divide evenly into 6 pieces (about 4 ounces each). Roll 1 dough piece into a 1/2-inch-thick rope (about 22 inches long), dusting liberally with flour as needed to prevent sticking. Cut rope crosswise into about 22 (1-inch-long) gnocchi pieces; press lightly with a fork or gnocchi paddle to create ridges. Transfer gnocchi to a lightly floured baking sheet. Repeat process with remaining 5 dough pieces. (Shaped gnocchi pieces can be used immediately or covered and chilled up to 8 hours or frozen up to 1 month.)
Bring 4 quarts water to a boil in a large saucepan over medium-high. Fill a bowl with ice water. Stir 1/4 cup salt into boiling water. Working in 2 batches, add gnocchi to boiling water; boil until gnocchi are tender and rise to surface, 2 to 3 minutes. Using a slotted spoon, transfer gnocchi to ice water; let stand 1 minute. Lightly grease a rimmed baking sheet with olive oil. Drain gnocchi, and transfer to prepared baking sheet. Chill, uncovered, at least 30 minutes or up to 4 hours.
Pat gnocchi dry. Heat a large skillet over medium. Add 1 tablespoon butter, swirling skillet until butter is melted and foamy. Add half of gnocchi (about 21/2 cups) in a single layer. Cook, undisturbed, until bottoms are light golden brown, 2 to 3 minutes. Add 3 tablespoons water, 1 1/2 tablespoons butter, 1 teaspoon vinegar, and 1/2 teaspoon salt. Cook, stirring and shaking skillet constantly, until butter melts and sauce is emulsified, 1 to 2 minutes. Stir in 2 teaspoons chives; season with salt to taste. Divide mixture evenly between 2 serving bowls. Drizzle each serving with 1/2 teaspoon nectar, and sprinkle each with about 1 tablespoon Parmesan. Repeat process using remaining ingredients. Serve immediately.