Prepare the Crescent Roll Base: Preheat your oven to 350 °F (175 °C). Open one package of crescent rolls and gently fit them into the bottom of an ungreased 9×13″ dish, pinching seams to create a solid crust. Letting the dough warm slightly helps avoid tears.
Build the Filling: In a saucepan over medium heat, melt the butter. Add both the cubed chicken and rotisserie chicken, then toss in the frozen mixed vegetables and drained potatoes. Stir frequently so everything heats through—this takes about 5 to 7 minutes. Warm the soups in a separate pan with the milk over medium-low heat, stirring until smooth and combined, about 3 minutes.
Combine Everything: Pour the creamy soup mixture into the chicken and veggies, stirring gently until well-coated. Season with salt and pepper, tasting to get the balance just right. Transfer the whole mixture into the baking dish over the crescent base, spreading it evenly.
Add the Crescent Top: Unroll the second package of crescent rolls and lay them over the top without pinching seams; this allows air to escape as they bake. Lightly tent the dish with foil to prevent over-browning of the crust during the longer bake time.
Bake and Rest: Bake uncovered for 40 to 50 minutes, until the filling is bubbly and the top crescent layer is golden walnut-brown. For a richer color, brush the top with melted butter or an egg wash after removing the foil halfway through. Let the pie sit for 10 to 15 minutes before cutting—this helps the filling thicken and makes servings cleaner.