Gradually add the milk and chicken broth mixture to the saucepan, starting with small amounts and whisking continuously. The sauce will thicken initially; keep adding the liquid slowly, whisking to maintain a smooth consistency. Once all the liquid is incorporated, raise the heat to medium and bring the mixture to a boil, still whisking. Stir in ½ teaspoon of salt and ⅛ teaspoon of pepper, allowing the sauce to boil for 1 minute before removing it from heat.