Chop and sauté the onion in a large skillet with the butter over low heat until translucent but not brown. Transfer half of the sautéed onions to a large mixing bowl.
Add the cubed pumpkin flesh and vegetable broth to the remaining onions in the skillet and cook for about 25 minutes, or until the pumpkin is very tender.
Cut the bread into small cubes and add to the bowl with the sautéed onions. Add the beaten eggs, milk, flour, chopped parsley, and a pinch of salt. Mix well with your hands to create a fairly uniform consistency. If needed, wet your hands to prevent the dough from sticking.
Bring a large pot of salted water to a boil.
Let the bread mixture rest for a few minutes. Meanwhile, flake the well-drained tuna with a fork and mix it with half of the grated cheese.
Shape the bread mixture into dumplings about the size of small walnuts. Create a small indent in the center of each ball, add a spoonful of the tuna mixture, then reshape the balls to seal the filling inside.
Cook the stuffed knödel in the boiling water for about 10 minutes, or until they float to the surface. Blend the pumpkin sauce until smooth and spoon onto four individual serving plates. Arrange equal numbers of knödel on each plate. Garnish with chopped chives and remaining grated cheese.