Heat a large sauté pan over medium-high heat. Swirl in the olive oil and sauté the diced onion until translucent. Add in the garlic and the chopped broccoli and sauté for 3 minutes, or until the broccoli is just starting to get tender but still bright green. Season with ground nutmeg, salt, and freshly ground black pepper, then drizzle in the fresh lemon juice. Remove from heat and let cool until no longer steaming. Stir in the whisked egg and feta cheese.
Preheat oven to 400°F (200°C). Take one sheet of filo dough and lay it on a clean surface. Brush completely with melted butter. Lay another sheet on top. Spoon 2 tablespoons of the broccoli filling 1" from the bottom edge, in the middle. Fold the left side of the filo sheet over, then fold the right side over. Fold the bottom edge towards the top and roll. Place on a baking sheet seam side down. Brush the top with more melted butter. Repeat with remaining sheets to make 10 rolls.
Bake for 20 minutes or until the tops are golden brown and crispy. The Broccoli Spanakopita should be golden and flaked beautifully.
