Remove and reserve the broccoli florets. Chop the stems.
Saute the onion in the butter in a large saucepan over medium heat for 5 minutes.
Add the chopped broccoli stems and potatoes. Cook for 5 minutes.
Stir in the flour. Cook for 5 minutes, stirring constantly. Stir in the chicken broth. Bring to a boil, stirring constantly and reduce the heat. Simmer for 15 minutes.
Process in small batches in the blender. Strain into a clean saucepan.
Cook the broccoli florets in boiling salted water to cover in a saucepan for 2 minutes. Drain and add to the soup.
Bring the cream just to a boil in a small saucepan. Stir in the nutmeg, salt, pepper, and 1/4 cup of the cheese. Cook just until the cheese melts (DO NOT BOIL). Stir into the soup. Spoon into 6 ovenproof soup bowls. Sprinkle with the remaining cheese.
Roll the puff pastry shells into circles 2 inches larger than the tops of the soup bowls. Brush the rims of the soup bowls with beaten egg. Place the pastry circles on the tops of the bowls and press to seal.
Place the bowls on a baking sheet. Bake at 400 degrees for 12 to 14 minutes or until golden brown.