1. Start with fully cooked chicken and thawed, cold puff pastry. Follow the pastry package for thawing; refrigerator thawing may take about 4 hours and is additional to the recipe time. Heat the oven to 400°F (205°C). Set two 16-fluid-ounce ramekins rated for this temperature on a rimmed baking sheet. Do not use storage lids in the oven.

2. Put the finely diced carrots and potatoes in a small saucepan and cover with lightly salted water. Bring to a boil, then simmer for 6–8 minutes, until nearly tender when pierced. Drain thoroughly.

3. Meanwhile, melt the butter in a small skillet over medium heat. Cook the onion for 2–3 minutes until softened. Stir in the flour and cook for 1 minute. Gradually whisk in the broth and milk until smooth. Add thyme and a little salt and pepper; simmer for 2–3 minutes until lightly thickened.

4. Stir in the cooked chicken, drained vegetables, and thawed peas. Heat for 1–2 minutes, stirring, then remove from the heat. Let stand for about 10 minutes, until warm rather than steaming vigorously. This cooling time is included in the total. Keep the pastry refrigerated while you wait.

5. Divide the filling between the ramekins, leaving about 1/2 inch below the rims; do not force in excess filling. On a lightly floured surface, unfold the pastry. If needed, roll gently just enough to cut two rounds about 1/2 inch wider than the outside ramekin rims, without rolling paper-thin. Use a full sheet as needed rather than assuming half will cover both dishes.

6. Lightly brush the ramekin rims with egg wash. Place a cold pastry round over each dish and press gently around the rim to secure. Cut a small cross or slit in the center of each lid. Brush the tops lightly with egg wash, avoiding heavy drips down the cut pastry edges. Discard unused egg wash.

7. Bake on the center rack for 25–30 minutes, checking from 20 minutes, until the pastry is well risen, deeply golden, and cooked through and the filling reaches 165°F (74°C) in the center of each pie. Insert an instant-read thermometer through the vent, avoiding the ramekin. If pastry browns early, tent loosely with foil and continue until the filling is hot enough. Timing varies with dish size, filling temperature, and oven.

8. Rest for 5 minutes. Toss lettuce, tomatoes, and red onion with the oil and lemon juice. Using oven mitts, set each hot ramekin on a heat-safe dinner plate with salad alongside. Garnish with parsley if desired and warn diners that the ramekins and filling remain hot.