Melt butter in a large skillet over medium-low heat. Add cabbage and cook until softened but not browned, about 30 minutes. Remove from heat and let cool, 10 to 15 minutes.
Preheat oven to 400 degrees F (200 degrees C).
Mix cooled cabbage with hard-boiled eggs and dill. Season with salt.
Line a pie plate with 1 sheet of puff pastry. Add cabbage filling, leaving some space around the edge. Cover with the second sheet of puff pastry and pinch edges together. Glaze the top with beaten egg.
Bake in the preheated oven until golden brown, 40 to 45 minutes.