- To cook your own chicken, half fill a large saucepan with water and bring to a boil. Add chicken. Cover and adjust heat so it simmers 30 minutes if using chicken pieces or 1 hour for a whole chicken. Drain and run under cold water to stop the cooking.
- While chicken cooks, leave skin on potato and slice in half. Microwave until it can be easily pierced with a fork, about 4 minutes. You want it tender but not soft. Peel off skins, then cut potato into bite-size pieces. If using carrots, peel and slice into ½ inch (1 cm) pieces. They should measure about 3 cups (750 mL) Boil in water until tender, 10 to15 minutes. Drain and set aside.
- Clean leeks. Slice in half lengthwise, then crosswise into 1 inch (2.5 cm) pieces. They should measure about 4 cups (1 L). Slice celery. You want about 2 cups (500 mL). Melt butter in a large wide saucepan over medium heat. Add leeks, separating pieces as you add. Stir in celery and stir often until starting to soften, about 10 minutes. If they begin browning, lower heat.
- Sprinkle with seasonings and salt, then flour. Stir constantly for 3 minutes, scraping the bottom of pan often. Stir in milk and continue stirring or whisking until no lumps remain. It will be gluey thick. Stir in coconut milk and continue stirring until fairly smooth. Stir in potatoes or carrots. Turn off the heat and cover.
- Remove skin and bones from chicken, then cut into large bite-size pieces. They should measure about 4 cups (1 L). Stir into the filling. If making the pie right away, stir gently over medium heat until chicken is hot. If making ahead, cover and refrigerator up to 2 days.
- Preheat oven to 375°F (190°C). The ideal baking dish is one where the filling comes right up to the top of the dish. This filling makes about 8 cups (2 L) which will fill most rectangular or oval dishes about 11 inches (27.5 cm) long. Turn filling into the dish and roughly smooth the top.