In a large saucepan, melt butter over medium-high. Add onion and carrots and cook until onion is translucent, about 4 minutes. Add flour and cook, stirring, 1 minute. Gradually add broth, stirring constantly, then bring to a boil. Reduce heat and simmer 5 minutes. Stir in chicken, peas and thyme and season with salt and pepper.

Place filling among in a 2-quart casserole dish. Roll out 1 sheet puff pastry to fid over casserole dish. Cut a small X's around crust to vent, then place on top of filling. Brush with beaten egg. Place dish on rimmed baking sheet. Bake at 375 degrees until filling is bubbling and crust is golden, 30 to 40 minutes.