Quarter and thinly slice the mushrooms. Heat the butter in a heavy saucepan over medium-high heat. Add the onion and mushrooms, season with salt and pepper, and cook, stirring occasionally, until softened and any liquid mushrooms give off has evaporated. Add broth and cream and simmer 15 minutes. In a blender, puree half of mixture (use caution when blending hot liquids). Stir puree and Sherry into the soup remaining in pan. Heat soup until hot, season with salt and pepper, and set aside to cool.
Divide the soup among 4 (1 1/2- to 2-cup) deep, oven-safe bowls about 5 to 6 inches in diameter, and stir some of the chives into each.
Using a pizza cutter, cut out discs from the puff pastry about 1/2-inch larger in diameter than the width of the soup bowls. Brush the inner edges of the pastry disks with some of the beaten egg. Invert the pastry discs (egg washed-side down) over the soup bowls and press the edges onto the sides to seal. Brush the tops with some of the beaten egg. Refrigerate for 1 hour or up to a day in advance.
When ready to serve: Preheat the oven to 400 degrees F. Put the soup bowls on a baking sheet and bake until golden brown, puffed, and steaming, about 30 minutes. Serve.