Unfold pastry and roll it out on a lightly floured work surface to ¼-inch thick. Use a festive cookie cutter to cut shapes in the dough, combining and rerolling scraps to use the whole thing. Transfer cutouts as you go to a parchment-lined baking sheet and refrigerate. Chill dough for 10-15 minutes should it become flabby, then continue to make and chill the remaining cutouts. Refrigerate until ready to use. Can be done up to 3 days in advance.

Heat 1 tablespoon butter in a large cast-iron skillet over medium heat until foaming. Add red onion and thyme. Stirring occasionally, cook until onion becomes translucent, about 5-7 minutes. Add radishes and carrots and cook mixture 5 minutes more, turning the veggies to brown evenly. Add the chicken and parsley. Season with hot sauce, salt, and pepper, and stir to combine. Remove from heat.

Preheat oven to 400°.

In a medium saucepan, add the remaining butter and melt until foaming over medium heat. Add flour and cook, whisking often, until mixture turns golden and nutty, about 2-3 minutes. Whisk in buttermilk, season with salt and pepper. Bring just to a boil. Reduce heat to simmer, whisking occasionally, until mixture is thick enough to coat a wooden spoon, 8-10 minutes.

Whisk in the broth. Pour béchamel-stock mixture onto pie ingredients and fold to incorporate. Transfer mixture into a medium enameled or heavy bottom skillet on a baking sheet.

Layer puff pastry cutouts onto filling like shingles. Gently press on those at the edge to seal. Brush entire pastry surface with beaten egg.

Bake for 35-40 minutes or until bubbling and deeply golden, rotating pan around the 20-minute mark. Cool on a wire rack for 5 minutes, then serve onto individual plates tableside.