Melt the butter in a small saucepan over medium heat. Whisk in the flour and cook, whisking constantly, until the mixture is bubbly and pale golden, about 2 minutes.

Remove from heat. Slowly pour in about a third of the cold milk, whisking until smooth. Add the remaining milk in a steady stream while whisking. Return to medium heat, whisking constantly, and bring to a gentle boil until the sauce thickens enough to coat the back of a spoon, about 4 to 5 minutes.

Reduce the heat to low and simmer for 2 minutes, whisking occasionally. Remove from heat, stir in the salt, white pepper, and nutmeg. Spoon over your dish right away.