For the pot pie filling: Melt the butter in a 1-quart saucepan over medium-low heat. Add the flour and whisk until smooth, 1 to 2 minutes. Gradually add the milk, whisking constantly to prevent lumps. Simmer over medium-low heat, whisking constantly, until the sauce is thick and smooth, about 6 minutes. Remove the pan from the heat and stir in the salt, pepper, Parmesan, chicken, and broccoli.