Heat the oil and butter in a pot over low heat. Add the onion; cook, stirring, for 10 minutes. Add the garlic; cook 2 minutes more. Sprinkle with the flour and tarragon; cook, stirring constantly, for 1 to 2 minutes more.
Add the carrots, potatoes, apple, and broth; bring to a boil. Reduce heat to a gentle simmer and cook, partially covered, until the vegetables are tender, about 20 minutes. Add the peas, dill, and chicken; season to taste with salt and pepper, then cook 5 minutes more.
Preheat the oven to 350°F.
Spoon the mixture into a 2-quart round ovenproof casserole. On a lightly floured surface, roll puff pastry out to form a circle 2 inches larger than the casserole. Lay the pastry over the top, trim the overhang to 1 inch, and crimp the edges around the rim to seal. Cut several slits in the pastry to release the steam, then brush with remaining egg wash. Bake until crust is golden, 40 to 45 minutes.