Chill the fat: Place the stick of butter for the biscuits in the freezer for 10 minutes. Meanwhile, position a rack in the center of the oven and heat to 425°F. Lightly grease a 12-inch oven-safe skillet (or a 9x13-inch baking dish) and set aside.
Make the biscuit dough: In a large bowl whisk together the flour, baking powder, and sea salt. Grate the chilled butter on the large holes of a box grater directly into the bowl, tossing to coat the shreds with flour. Drizzle in the cold buttermilk and stir with a fork just until the dough comes together and no dry pockets remain.
Laminate quickly for flakes: Turn the dough onto a lightly floured surface. Pat into a 1/2-inch-thick rectangle, fold it in thirds like a letter, rotate, and repeat 2 more times. Pat to about 3/4-inch thick and cut into 12 squares with a sharp knife (no twisting). Transfer the cut biscuits to a plate and freeze while you start the filling.
Warm the liquids: In a microwave-safe measuring cup, heat the chicken stock and milk together until warm, 1–2 minutes. This helps the sauce thicken smoothly.
Build the filling base: Melt the butter in the prepared oven-safe skillet over medium heat. Add onion, celery, 1/2 teaspoon of the kosher salt, and cook until tender and lightly translucent, 6–7 minutes. Stir in the garlic and thyme and cook 30 seconds until fragrant.
Thicken the sauce: Sprinkle the flour over the vegetables and stir constantly for 1–2 minutes. Slowly whisk in the warm stock and milk, scraping the skillet to release any browned bits. Stir in the Dijon. Bring to a gentle simmer and cook until glossy and slightly thickened, about 4 minutes.
Finish the filling: Stir in the frozen mixed vegetables and cook 3 minutes. Fold in the shredded chicken, the remaining 1 teaspoon kosher salt, and the black pepper. Taste and adjust seasoning if needed. If not using a skillet, transfer the filling to the prepared baking dish now.
Top and bake: Arrange the chilled biscuits evenly over the hot filling. Bake at 425°F for 12 minutes, then reduce the oven temperature to 375°F and continue baking 15–18 minutes, until the biscuits are deep golden and the filling is bubbling around the edges.
Finish and rest: Brush the biscuit tops with melted butter during the last 5 minutes of baking. Let the pot pie rest 10 minutes before serving to allow the gravy to settle.
Tip: For crisper biscuit bottoms, keep the biscuits very cold and make sure the filling is hot when you top it—this jump-starts steam and lift for flakier layers.