Make pie crust. Place flour and salt in a food processor and pulse to combine. Scatter chilled butter pieces on top. Pulse 10 to 12 times or until mixture resembles coarse crumbs. Add cheese and pulse to combine.: You should notice a mixture that looks like coarse sand with visible pea sized bits of butter , and the sound of the processor will shift from clumpy to more even as the pulses continue. This texture ensures pockets of steam during baking, resulting in flaky layers. The aroma will turn faintly dairy rich when the cheddar cheese is incorporated, adding a buttery, sharp note. Common mistakes include overprocessing which can make the dough mealy and dense, or using warm butter that smears instead of remaining distinct. If your mixture looks too uniform, pulse fewer times to restore texture.
Transfer mixture to a large bowl. Sprinkle 5 tablespoons of ice water on top. Stir and press the dough together using a rubber spatula. If dough is not coming together, use a little more ice water.: As you work the dough, you will feel it gradually cohere into a shaggy mass, cool and slightly tacky. The cold water activates just enough gluten to bind while keeping the butter chilled. The tactile feedback is important, you should stop when it holds together without feeling wet. Too much water makes the dough sticky and elastic, which yields tough crust; too little and it will crumble. If you need more water, add it a teaspoon at a time, avoiding overhydration.
Turn dough out onto a piece of plastic wrap. Flatten into a disk, wrap well and refrigerate for at least 1 hour and up to 24 hours.: After wrapping, the dough will firm and the chilled butter will solidify, which keeps the layers distinct when baked. Cooling also relaxes the gluten so the dough is easier to roll and less likely to shrink. You may feel the disk cool to the touch; that is a good sign. Common issues include skipping the chill step which often causes tearing or excessive shrinkage during baking, so be patient and allow the dough time to rest.
Preheat oven to 400 degrees.: You will sense the kitchen warming as the oven comes to temperature, and preheating ensures the crust starts baking immediately on contact, producing flake and lift. A properly preheated oven also helps the filling to bubble at the same time the crust browns. Avoid placing the pie in an oven that has not reached 400 degrees, as that can produce a soggy bottom and an underbaked crust.
To make filling, melt butter in a medium (10-inch) cast iron skillet over medium heat. Add onion, carrot, and celery and cook 5 minutes.: As the butter melts and the vegetables sizzle, you'll smell a sweet, savory aroma develop. The vegetables should soften and become translucent, releasing natural sugars that deepen the filling's flavor. A steady, moderate sizzle is ideal, not a loud pop which signals too high heat. If the vegetables begin to brown aggressively, lower the heat to prevent bitterness.
Sprinkle flour on top. Cook and stir for 1 minute.: The flour should absorb the melted butter and coat the vegetables, forming the start of a roux. You will hear a gentle thickening sound and see the mixture become slightly glossy. Cooking the flour removes the raw taste and sets the stage for a smooth sauce when liquids are added. Do not skip this brief cooking time; undercooked flour can leave a pasty mouthfeel in the finished filling.
Gradually add chicken broth whisking it into the flour well. Whisk in half and half.: Add the chicken broth slowly while whisking so the mixture remains lump free. The sauce will thicken and take on a velvety sheen as the starches swell. When you pour in the half and half , watch the sauce smooth out and become creamy, and listen for a gentle simmer. If you add the liquid too quickly, you may get lumps; keep whisking and if lumps form, strain or whisk vigorously until smooth.
Add salt, pepper, thyme, and poultry seasoning. Let mixture simmer for a few minutes.: The simmer concentrates flavors and reduces the sauce slightly, giving it body. You will smell the dried thyme and the seasoning blend blooming, and the texture should feel thicker on the back of a spoon. Simmer only until the sauce coats the spoon; prolonged simmering can thin the filling as liquid reduces excessively or cause the dairy to separate.
Stir in peas and chicken. Remove from heat.: Folding in the defrosted peas and diced cooked chicken finishes the filling. The contrast of warm gravy and tender morsels is key, and the green pops of peas brighten the plate. Remove from heat promptly to avoid overcooking the peas, which can turn them mushy. A common error is adding cold chicken straight from the fridge which cools the sauce; bring chicken closer to room temperature first.
Remove dough from refrigerator and let it sit on the counter for 10 minutes.: This brief rest makes the dough more pliable for rolling while still keeping the butter cool. You will notice it soften slightly at the edges, which reduces crack risk when rolling. Do not let it sit too long or the butter will warm, making the dough sticky and harder to lift into the skillet.
Sprinkle some flour on a piece of parchment paper. Place dough on paper and roll it into a circle that’s about 1/8-inch thick. Slide your hands under the parchment paper and transfer the pie dough to the pan, flipping it over on top of the pie filling and peeling the paper off of it. Trim excess dough or tuck it under the so that the crust fits the pan. Brush top with egg. Cut a few slits in the dough to allow steam to escape.: Rolling on parchment keeps the dough from sticking and helps you transfer it intact. The crust should look even and slightly translucent where it thins, and the sound of the rolling should be smooth and quiet. Brushing with the beaten egg adds sheen and encourages a deep golden color. Forgetting to cut vents traps steam and can cause soggy pockets, so be sure to make slits.
Place in oven and bake 30 to 35 minutes, until pie crust is lightly browned and no longer feels doughy.: While baking you'll see the filling bubbling gently at the edges and the crust turning a rich golden tone. The aroma will be rich and buttery with a toasted cheddar cheese note from the crust. Check for a firm, set filling and a crust that feels cooked rather than doughy. Removing too early risks a runny center, while overbaking can dry the filling and make the crust overly crisp. Let it rest briefly before serving so the gravy settles.