How to Make It
Add water to bottom of a steamer pot to a depth of 1/2 inch; bring to a boil over medium-high. Place steamer basket in pot. Add beets; cover and reduce heat to medium. Steam until beets are tender when pierced with a knife, about 20 minutes.
Transfer beets to a blender. Add 2 tablespoons butter, 2 tablespoons vinegar, 2 tablespoons water, 1/2 teaspoon salt, and preserved lemon. Process until smooth, about 1 minute, stopping to scrape down sides as needed. Transfer beet puree to a small bowl, and set aside.
Pierce 1 pound sweet potatoes all over with a fork, and wrap in damp paper towels. Microwave on HIGH until tender, about 10 minutes, turning after 5 minutes. Let cool slightly. Cut potatoes in half; scoop out pulp to equal 3/4 packed cup; discard skins. Place pulp in a medium bowl; mash, and set aside.
Peel remaining 1/2 pound potatoes, and cut into 1/4-inch cubes to equal 1 cup. (Reserve any remaining sweet potato for another use.) Heat remaining 1 tablespoon butter in a large skillet over medium until sizzling. Add cubed sweet potatoes; cook, stirring often, just until tender but not browned, about 5 minutes. Transfer 1/3 cup cooked sweet potato cubes to bowl with mashed sweet potatoes. Stir in hazelnut butter, sumac or lemon juice, coriander, pepper, and 1 1/4 teaspoons salt until thoroughly blended. (You should have about 1 cup sweet potato filling.)
Toss together cilantro, remaining cubed sweet potatoes, remaining 1/2 teaspoon rice vinegar, and remaining 1/4 teaspoon salt; set aside.
Working with 1 wrapper at a time, brush edges of wrapper with water. Spoon 2 teaspoons sweet potato filling into center of wrapper. Fold wrapper in half over filling; press and pleat edges to seal. Transfer to a baking sheet; cover with a clean, dry kitchen towel. Repeat procedure with remaining wrappers and filling.
Heat 1 tablespoon oil in a large nonstick skillet over medium. Arrange half of dumplings in a single layer in skillet. Cook, undisturbed, until bottoms are golden brown, about 5 minutes. Add 1/4 cup water, cover, and increase heat to high. Cook until water is mostly evaporated and wrappers are cooked through, about 3 minutes. Transfer to a plate, and cover loosely with aluminum foil to keep warm. Repeat procedure with remaining 1 tablespoon oil, remaining dumplings, and remaining 1/4 cup water.
Spoon 2 tablespoons beet puree into each of 6 shallow bowls; top with 4 gyoza. Sprinkle gyoza evenly with cubed sweet potato mixture and sliced scallions.