> Melt butter in a saucepan over medium heat. Whisk in flour until it forms a loose mixture, not a paste.
>Gradually pour milk into butter-flour mixture, whisking thoroughly after each addition to ensure no lumps. Simmer until thickened, about 5-7 minutes. The sauce should coat the spoon without running off, as shown in the photo below.
> Turn off heat and season béchamel with salt and pepper. Add eggs and Parmesan cheese, whisk until smooth with no lumps.