In a dutch oven, heat the butter and olive oil over medium heat. Pat the chicken thighs dry all over and season on both sides with salt and pepper. Add the thighs to the pot and brown undisturbed for 3 minutes on each side. Cover, reduce heat to low, and cook 4-5 minutes more, or until cooked through (the internal temperature should be 165F.) Leaving the drippings in the pan, transfer the chicken thighs to a cutting board and allow to cool. When cool enough to handle, dice the chicken and set aside.
Increase the dutch oven heat to medium. Add the shallots, celery, and onion, and cook, stirring occasionally, until onion is translucent, about 10 minutes. Sprinkle the spices, flour, and ½ tsp salt over the top and cook, stirring, about 2 minutes.
Add the chicken stock, potato, and butternut squash and bring to a boil. Boil 10 minutes, stirring occasionally. If bits stick to the bottom of the pot, don’t fret, this is normal.
Reduce the heat to a simmer and stir in the thyme, sage, milk, and cream. Taste for salt and adjust if needed. Simmer 15 minutes more, adding the peas for the last 2 minutes of simmering.
While the filling simmers, prepare the puff pastry. Unfold the pastry on a piece of parchment. Invert desired bowls or croques onto the pastry and use a sharp paring knife to trace and cut to size. Use a small cookie cutter to cut shapes out of the excess pastry if desired. Line two baking sheets with parchment and divide the pastry between them, leaving 1-2 inches between pieces of pastry. In a small dish, whisk together the egg and the remaining 1 tsp milk. Use a pastry brush to brush the top of the pastry pieces. Bake the pastry until puffed and golden.
When ready to serve, stir the diced chicken thighs into the filling. Ladle the filling into dishes and top with rounds of puff pastry. Serve with extra puff pastry cut outs for dipping!