Preheat + start the base: Preheat oven to 400 Fahrenheit (200 Celsius). In an oven-safe cast-iron skillet over medium heat, melt 3 tablespoons butter. Add onion (1 small), celery (2 stalks), and dried sage (½ teaspoon) with ½ teaspoon salt and ¼ teaspoon pepper. Cook until translucent and softened, 3–5 minutes.
Hard veg + mushrooms (bump the heat): Add carrots (2, diced), mushrooms (8 ounces), and thyme leaves (from 3 sprigs). Increase heat to medium-high and cook, stirring, until the mushrooms release their liquid, it evaporates, and they begin to brown, 5–7 minutes. Season with another ¼ teaspoon salt and ¼ teaspoon pepper.
Stir in garlic (2 cloves, minced) and cook 30 seconds until fragrant.
Roux right on the veg: Sprinkle ¼ cup all-purpose flour evenly over the skillet and stir 1–2 minutes to coat and cook off the raw flour taste.
Build the sauce: Gradually whisk in 2 cups chicken broth. Bring to a gentle simmer and cook, stirring, until thick and glossy, 5–10 minutes. Stir in paprika (½ teaspoon) and season with another ½ teaspoon salt and ¼ teaspoon pepper (adjust later to taste).
Finish the filling off heat: Remove from heat. Stir in ½ cup cream, 3 cups shredded chicken, and ½ cup peas. Taste and adjust salt/pepper (use the remaining ¼ teaspoon salt and ¾ teaspoon pepper only if you want it extra peppery).
Top, vent, brush: Drape puff pastry over the skillet, trim or fold the edges as you like, cut 3–4 vents, and brush with 1 beaten egg.
Bake: Set the skillet on a sheet pan and bake until the pastry is deep golden and the filling is bubbling at the edges, 20–25 minutes. Rest 10 minutes before serving.