Position rack in center of oven; preheat to 400F. 

Heat butter in a large Enameled Cast Iron Skillet or other deep, oven-safe skillet, over medium. Add flour and cook, using a silicone whisk or wooden spoon to stir constantly, until roux is foaming and light blonde in color, adjusting heat if necessary, about 3 minutes.

Add  carrots, celery, onion, garlic, 1 tsp. salt, and ½ tsp. pepper and cook, stirring, until tender and soft (but not browned), and roux just begins to stick to bottom of Skillet , 5–7 minutes.

Slowly add broth, 1 cup cream, 2 tsp. thyme, paprika,  nutmeg, 1 ½ tsp. salt, and 1 tsp. pepper, whisking to incorporate. Bring to a low boil to activate flour, then reduce heat and simmer until thickened, about 5 minutes. Taste sauce and adjust seasoning, if necessary.

Add chicken and potato and cook, stirring to incorporate. If you feel like your sauce is getting too thick, now is the time to add a little more chicken broth. Remove from heat and let cool 10 minutes while you prepare pastry.

Beat egg with remaining 1 Tbsp. cream to create an egg wash.

Lightly dust work surface with flour and unfold puff pastry. If your package has 2 sheets, place them side by side with a slight overlap and roll a floured Rolling Pin over to seal them together to create 1 big sheet. Use Rolling Pin to roll puff pastry into a 14" square (large enough to cover Skillet with a bit of overhang). Brush outside rim of Skillet with egg wash to help pastry stick. Roll pastry around Rolling Pin and unfurl over Skillet, draping it over top and allowing it to slump in a little bit without stretching.

Trim pastry using a Paring Knife so that there is a 1" border all around. Press pastry onto top and outside rim of Skillet. Reserve puff pastry trimmings.

Using a pastry brush, brush top of pastry with egg wash. Cut 5  small slits in center so steam can escape. Sprinkle top with  flaky salt, pepper, and remaining 1 tsp. thyme. Place on a Sheet Pan to catch any drips, and bake pot pie until crust is light golden brown, 30–35 minutes.

Reduce oven temperature to 350F and continue to bake until filling is bubbling around edges and crust is well browned, 10-15 minutes longer. Let sit 10 minutes before serving.

Slice into crust with a knife or use a large spoon to scoop out and plate filling with crust on top. Finish with parsley, if desired.

For an extra treat, cut puff pastry trimmings into any shapes you desire, then brush with remaining egg wash and sprinkle with cinnamon-sugar. Place on a Sheet Pan and bake alongside the pot pie until puffed and golden brown, about 15 minutes.