Put goose in a large pot and cover with water. Bring to a boil and skim fat from surface. Add mushrooms, Brussels sprouts, carrots, onions, celery, garlic, parsley sprigs, and turnip and season with salt and pepper. Reduce heat to medium-low and simmer, adding a little water, as necessary, to keep goose covered and skimming fat from surface, for 3 hours.
Set a fine strainer over a large pot and strain stock, reserving liquid and solids. Transfer goose parts to a platter; let cool. Transfer mushrooms, Brussels sprouts, and carrots to a plate, discarding the other vegetables, and let cool. Shred goose meat, discarding skin and bones, and add to stock. Julienne the reserved vegetables; return to stock. Season soup with salt and pepper; set over low heat.
Meanwhile, make the matzo balls: Put matzo in a food processor and pulse until coarsely ground; set aside. Beat together schmaltz and eggs in a large bowl; add baking powder and ginger; season with salt and pepper. Add reserved ground matzo and remaining parsley; mix vigorously with a wooden spoon until stiff; cover with plastic wrap; refrigerate for 30 minutes.
Bring a large pot of lightly salted water to a boil. Using wet hands, divide matzo mixture into 24 portions and roll into balls about 1" in diameter. Working in 2 batches, add matzo balls to pot, bring water to a boil, and reduce heat to medium. Cover and simmer until matzo balls are tender and doubled in size, about 20 minutes. To serve, divide soup and matzo balls between 12 bowls.