Step 1

Preheat oven to 400°. Heat butter in a large skillet over medium heat. Add shallot and thyme and cook, stirring often, until shallot is translucent and golden, about 4 minutes. Add ½ cup flour and cook, whisking constantly, until mixture is pale golden and smells like popcorn, about 5 minutes.

Step 2

Whisk in broth ½-cupful at a time, incorporating completely before adding more. Whisk in cream; season with salt and pepper. Bring mixture to a boil, reduce heat, and simmer, whisking occasionally, until thick enough to coat a spoon, about 3 minutes. Reduce heat as needed to maintain a simmer and add carrots; cook until nearly tender, about 5 minutes.

Step 3

Mix in chicken, pearl onions, and peas, then add hot sauce, Worcestershire sauce, and vinegar and season with salt and pepper.

Step 4

Transfer mixture to a shallow 2-qt. baking dish. Unfold pastry and gently roll out on a lightly floured work surface just to smooth any creases and make large enough to fit over dish, if needed. Drape over baking dish and trim pastry so it is slightly larger than the dish if using puff pastry (you want a little overhang), or crimp dough around edge of dish if using pie dough. Brush with egg and make 2 or 3 slits in the pastry with a sharp knife.

Step 5

Place baking dish on a rimmed baking sheet (to catch any filling that may drip over) and bake until puff pastry is golden brown and filling is bubbling through slits in pastry, 20–30 minutes. Reduce oven temperature to 350° and continue to bake until crust is deep golden brown and cooked through, 30–35 minutes longer. Let sit 10 minutes before serving.