Heat the oven to 400°F. In a small saucepan or microwave-safe cup, gently warm the milk until steaming but not boiling; keep it nearby (warm milk helps the sauce stay silky).
Start the filling: In a wide skillet over medium heat, melt 2 tablespoons of the butter. Add the onion and carrot with a pinch of salt. Cook, stirring often, until the onions are translucent and the carrots soften slightly, 5 to 6 minutes.
Add the remaining 1 tablespoon butter and the mushrooms. Increase heat to medium-high and cook until the mushrooms release their liquid and take on some color, 6 to 8 minutes. Stir in the black pepper and poultry seasoning.
Sprinkle the flour evenly over the vegetables and cook, stirring, for 1 to 2 minutes to form a roux. Pour in the dry sherry and let it bubble for 30 seconds, scraping the pan.
Gradually whisk in the warm milk and add the kosher salt. Bring to a gentle simmer and cook, stirring, until the sauce is smooth and slightly thick, about 3 to 5 minutes.
Fold in the peas, pimientos, and shredded chicken. Simmer 2 minutes more, then take off the heat, cover, and let the mixture rest 5 minutes to thicken. Stir in half the parsley and keep warm on low.
While the sauce cooks, arrange the frozen puff pastry shells on a parchment-lined baking sheet. Bake at 400°F until well risen and deeply golden, 22 to 25 minutes. Let stand 5 minutes, then lift off the caps and gently remove any soft centers to make room for the filling.
Spoon the hot chicken mixture into the warm pastry shells. Top with the caps and sprinkle with the remaining parsley. Serve at once.