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Stirring flour into softened vegetables for exactly ten minutes cooks out the raw starch flavor before liquid is added. Building this base requires a wide, heavy-bottomed Dutch oven or deep skillet that allows even heat distribution across the butter and aromatics. Stainless steel or enameled cast iron works well for this stage, preventing hot spots that might scorch the dairy later. Dice the potatoes and carrots uniformly to ensure they soften evenly during the initial ten-minute sauté.
Gradually whisking the combined half-and-half and chicken broth into the flour-vegetable mixture prevents lumps from forming in the sauce. Maintain medium heat while stirring constantly until the liquid reaches a bubbling simmer and thickens visibly. Pour the finished filling into a standard 9x13-inch casserole dish before draping the rolled puff pastry across the top. Cut sharp slits into the pastry surface to allow steam to escape efficiently during the baking phase at 400 degrees Fahrenheit.