Make the crust first. Mix flour and salt in a bowl. Add cold butter cubes and use a pastry cutter or fork to break them into pea-sized pieces. Add cold water a tablespoon at a time, mixing gently until dough just comes together. Don't overwork it — lumpy is better than smooth here. Wrap in plastic and chill for 20 minutes minimum.
While dough chills, prep your vegetables. Dice potatoes small (they cook faster), mince garlic, and measure everything out. Having ingredients ready makes the filling come together in minutes. This is also when you'll shred your chicken if using a whole bird.
Heat a large skillet over medium heat. Add 2 tablespoons butter and sauté garlic for 30 seconds until fragrant. Add diced potatoes and carrots, cooking for about 5 minutes until they start to soften. You'll hear them sizzle as they hit the pan — that's your cue they're cooking properly.
Sprinkle 3 tablespoons flour over the vegetables and stir constantly for 2 minutes. This creates your roux and prevents lumps later. Add chicken broth slowly, stirring the whole time so no clumps form. The mixture should look like thick soup, not watery.
Add heavy cream, thyme, shredded chicken, and peas. Stir everything together and let it simmer for 3 minutes. Taste and add salt and pepper as needed. I always add a pinch more thyme because dried herbs need that boost.
Pour the filling into a 9-inch pie dish. When your kitchen smells like warm, savory chicken and herbs mixing together, you know the filling's ready. Let it cool for 5 minutes while you prep the crust. This prevents the hot filling from making the bottom crust soggy.
Roll chilled dough between two pieces of parchment paper until it's about 1/8-inch thick and slightly larger than your pie dish. Transfer to the dish and crimp edges with a fork. Bake at 425°F for 8 minutes, then lower heat to 375°F and bake another 35-40 minutes until crust is golden brown.