01 - Set oven temperature to 400°F (204°C) and allow to fully preheat.
02 - Melt butter in a large skillet over medium heat. Add diced onion, minced garlic, carrots, and celery. Cook, stirring occasionally, for 5 minutes until softened.
03 - Sprinkle flour evenly over vegetables and stir constantly for 1 to 2 minutes to eliminate raw flour taste.
04 - Gradually pour in chicken broth and whole milk, whisking to prevent lumps. Simmer 3 to 4 minutes until mixture thickens.
05 - Stir in shredded chicken breast, frozen peas, salt, black pepper, dried thyme, and sage. Continue cooking for 2 to 3 minutes until heated through. Remove from heat.
06 - Evenly transfer the filling into a 9-inch pie dish.
07 - On a lightly floured work surface, roll out puff pastry sheet to fit pie dish. Slice pastry into 1-inch wide strips using a sharp knife or pizza cutter.
08 - Arrange pastry strips over filling in a crisscross manner, leaving occasional gaps for a 'bandaged' visual effect.
09 - Place two sliced black olives at one end of the pie to mimic eyes.
10 - Lightly brush the surface of pastry strips with the beaten egg to promote browning.
11 - Bake in preheated oven for 35 to 40 minutes until pastry turns golden and crisp. Allow to rest 10 minutes before serving.