Melt butter in large saucepan over low.
Add flour, and slowly whisk until it resembles wet sand.
Cook over low, whisking constantly, until mixture is shiny and smells toasty, about 2 minutes.
Whisking constantly, gradually add cold milk.
Once fully combined, switch to a wooden spoon.
Add onion half, salt, and nutmeg.
Cook over low to medium-low at a very gentle simmer, stirring very often, until onion is cooked through and sauce is very thick, about 45 minutes.
Pour through a fine wire-mesh strainer; discard solids.
Use warm, or refrigerate as directed.
