Place a rack in center of oven; preheat to 400F
Remove and discard skin of 1 rotisserie chicken, then shred meat into 1“ pieces (you should have 4–5 cups meat); set aside
Finely dice onion and place in bowl
Peel turnips and cut into 3/4 inch pieces. Coarsely chop garlic. Combine turnips, garlic, and thyme into a separate bowl
Melt butter in a 12” cast-iron skillet over medium-high. Add onions and cook until browned and soft, about 20 minutes. Add turnips, garlic, and thyme along with 1 tsp salt and 1/2 tbsp pepper. Cook for about 5 minutes, until turnip is softening
Sprinkle flour over vegetables and cook for about 30 seconds, until flour begins to stick to the bottom of the pan
Add wine and continue cooking for a couple of minutes
Set aside 1 tbsp of cream and add the rest to the pan along with the peas, the reserved chicken, and 1/2 tbsp salt. Cook, stirring constantly, for another couple of minutes. Taste and adjust seasoning. Remove from heat and transfer skillet to a baking sheet
Roll out puff pastry on a lightly floured surface into a 13“ square (large enough to cover skillet with a bit of overhang). Roll pastry up onto rolling pin. Unfurl pastry from rolling pin, draping it over skillet
Trim pastry so that there is a 1” border all around. Fold edge of puff pastry under itself. Crimp edges with a fork (just like you would do when making the top crust of a pie)
Using a pastry brush, brush top of pastry with reserved cream. Cut 5–6 small slits in the center so steam can escape. Bake pot pie until crust is light golden brown, 22–24 minutes
Reduce oven temperature to 350F and continue to bake until filling is bubbling around the edges and crust is well browned, 25–35 minutes longer. Let sit 10 minutes before serving