Melt the butter in a large oven-safe skillet or Dutch oven over medium heat. Add the onion, carrots, and celery with a good pinch of salt and cook, stirring now and then, until softened and the onion is translucent, about 7-8 minutes. Stir in the garlic and thyme and cook 1 more minute.

Sprinkle the flour over the vegetables and stir constantly for 1-2 minutes to cook out the raw-flour taste. You want a blond paste coating everything, not browned.

Pour in the warm broth a little at a time, whisking after each addition so it stays smooth, then whisk in the milk. Bring to a gentle simmer and let it cook, stirring, until the sauce is thick and glossy and coats the back of a spoon so a finger leaves a clean trail, 5-8 minutes. This is the whole game. Do not rush it off the heat while it is still thin.

Turn off the heat. Stir in the chicken, peas, Dijon, and lemon juice. Taste and season aggressively with salt and pepper. The filling should be thick enough to mound on a spoon. If you have time, let it cool for 15-20 minutes (or chill it) so it firms up even more before the crust goes on.

Heat the oven to 400F and slide a sheet pan onto the bottom rack to preheat. Scrape the filling into your baking dish (or keep it in the oven-safe skillet). Lay the chilled crust over the top, trim the overhang, crimp the edges, and cut 3-4 vents in the center. Brush all over with the beaten egg and sprinkle with flaky salt.

Set the pie on the preheated sheet pan and bake until the crust is deep golden brown and the filling is bubbling up through the vents, 40-45 minutes. If the edges brown too fast, tent them with foil.

Here is the part nobody wants to hear: let it rest 15 minutes before you cut into it. That rest lets the filling set so it scoops up thick instead of running all over the plate.