Dice the onion, carrots, and celery into small, even pieces so they cook at the same rate. Shred or dice your cooked chicken and set aside. Preheat oven to 400°F.

Melt butter in a large oven-safe skillet over medium heat. Add onion, carrots, and celery with a pinch of salt. Cook for 8-10 minutes, stirring occasionally, until vegetables soften and the onion turns translucent. The carrots should yield slightly when pierced with a fork.

Sprinkle flour over the vegetables and stir constantly for 1 minute to cook out the raw taste. Slowly pour in chicken broth while whisking to prevent lumps, then add milk and thyme. Bring to a gentle simmer and cook for 5-7 minutes, stirring occasionally, until the sauce thickens enough to coat the back of a spoon.

Stir in the shredded chicken and frozen peas. Taste and season with salt and black pepper as needed. The filling should be creamy and thick but still pourable; it will thicken more in the oven. If using a baking dish instead of an oven-safe skillet, transfer the filling now.

Arrange biscuit dough rounds on top of the hot filling, spacing them evenly. They will expand as they bake. Transfer to the oven and bake for 18-22 minutes, until the biscuits are deep golden brown on top and cooked through underneath. A toothpick inserted into a biscuit should come out clean.

Let the pot pie rest for 5 minutes before serving. This allows the sauce to set slightly so it does not run all over the plate. Serve directly from the skillet or baking dish.