Rach's tip: Put the frozen puff pastry in the refrigerator a few hours before you start cooking to give it time to thaw out.
Preheat the oven to 400°.
Line a baking sheet with parchment paper. Lightly flour a work surface and rolling pin. Unfold the pastry. Gently roll out the pastry a bit to smooth out the creases. Cut the pastry horizontally in half, then cut each half crosswise in half. Cut each section into 2 to 3 strips and arrange on the parchment paper. In a small bowl, beat the egg with about 2 tsp. water. Brush the pastry with the egg wash. Bake the puff pastry strips until puffed and deep golden, about 20 minutes.
Pull or chop the chicken meat into bite-size pieces. (Reserve the skin and bones for stock.)
Meanwhile, in a deep skillet, melt the butter over medium to medium-high heat. Add the carrot, fennel, shallot, and celery and cook, stirring often, until softened, about 5 minutes. Add the asparagus, garlic, and herbes de Provence; season with salt and pepper. Cook until the asparagus starts to soften, about 3 minutes more. Stir in 2 tbsp. flour. Whisk in the wine and cook until absorbed, about 30 seconds. Add the stock and the demi-glace (if using) and stir until dissolved. Cook, stirring often, until the sauce thickens, about 2 minutes. Stir in the cream and reduce heat to low. Add the tarragon, Dijon, and nutmeg. Stir in the chicken and peas. Cook until the chicken is heated through, about 2 minutes.
Transfer the chicken filling to a 3-quart casserole dish. Top with the puff pastry strips.