Dissolve the sugar and ginger in the lukewarm water, and sprinkle the yeast over it. Let it stand for 10 minutes. Combine the yeast with the lukewarm milk, eggs, and salt. Stir in the flour, and knead it until smooth. The dough should be soft. Cover and let it rise in a warm place until double in bulk.
Cut the dough into 2 parts for easier handling. Roll each part into a long roll about 1 1/2 inches in diameter. Cut the roll into 1/2 inch pieces. Flatten each piece and wrap it around a plum filled with sugar and cinnamon. Seal the edges securely. Place them on a floured board, cover, and let them rise.
Drop a few varenyky at a time into rapidly boiling salted water. Stir gently with a wooden spoon. When the varenyky rise to the top, cook them for about 2 minutes by rapid boiling.
Remove them with a perforated spoon to a colander and drain well. Do not overcook because this dough has a tendency to become somewhat firm.
Place the varenyky in a serving dish, sprinkle generously with melted butter, and toss lightly to coat them evenly. Serve hot with a sprinkling of sugar and browned buttered bread crumbs, or sour cream.