Preheat oven to 400°F (200°C). Set 6–8 (6–8 oz) ramekins on a rimmed sheet pan. Heat butter and olive oil over medium. Sauté onion, carrots, and celery with a pinch of salt until tender, 6–8 minutes. Stir in garlic and seasonings; cook 30 seconds.
Sprinkle flour over veggies; cook 1 minute. Slowly whisk in warm broth until smooth, then whisk in milk/cream. Simmer 2–3 minutes until thick and glossy. Turn off heat; fold in chicken and peas.
Spoon hot filling into ramekins, about ¾ full.
Cut rounds of puff pastry or pie crust slightly larger than ramekins; lay over tops and press to seal on the rim. Cut 2–3 small slits to vent. (For biscuits, place one biscuit on each; gently stretch to cover.) Brush tops with egg wash.
Puff pastry/pie crust — 20–25 minutes until deep golden and bubbling. Biscuits — 22–28 minutes until cooked through. Tent with foil if browning too fast. Cool 5–10 minutes before serving.