Make the Chicken: Season the chicken with salt and pepper.
Using scissors, round off the 4 corners of the parchment paper squares to make large circles (don't worry about the circle being perfect). Arrange each breast, slightly off center, on the paper circles, with the smooth rounded sides up. Fold one end of the paper over each breast, to make half-moon shapes. (The breasts should be horizontal to the folds.) Make 1/4-inch folds all around the open ends and press to crimp and seal. Close the final crimp with a paper clip. Place packages on a baking sheet and refrigerate until ready to use.
Make the Sauce: Heat the oil in a saucepot over medium heat. Add the shallot and garlic and cook, stirring, until soft, about 3 minutes. Add the roasted squash and fennel and season with salt, to taste. Cook, scraping the bottom of the pot with the wooden spoon to loosen any bits, about 3 to 4 minutes. Raise the heat to high, add the sherry, and cook, stirring, until completely evaporated. Add the chicken stock, bring to a boil, reduce the heat to a simmer, and cook for 7 minutes. Season with pepper. Remove from the heat and let cool.
Puree the sauce in a blender and strain it through a fine-mesh strainer lined with cheesecloth. (The sauce and can be made a day in advance.)
When ready to serve: Arrange a rack in the center of the oven and preheat to 400 degrees F.
Bake the chicken packages until puffed and lightly browned, about 20 minutes.
Heat the sesame oil in a skillet over medium-high heat. Add the bok choy, season with salt, and cook until lightly browned, about 2 minutes. Add the soy sauce and chicken stock, cover, and cook until soft but still firm in the center, about 5 minutes. Remove from the heat.
Meanwhile, reheat the sauce in a small saucepan and season with salt and pepper, to taste.
Divide the chicken packages among 4 plates. Using scissors, cut an "X" in the top of each package and pull back the edges to reveal the chicken. Spoon 2 to 4 tablespoons of the sauce over the chicken, and nestle 2 halves of the bok choy beside the chicken. Garnish with the cilantro sprigs and serve immediately.