Melt butter in a heavy saucepan over medium heat; whisk in flour until smooth. Gradually whisk in milk, salt, and red pepper.
Bring mixture to a boil over medium-high heat, whisking constantly. Boil, whisking constantly, 1 minute. Remove from heat, and stir in cheese until melted and blended.
Beat eggs with a fork until thick and pale. Gradually stir about one-fourth hot milk mixture into eggs; add egg mixture to remaining hot milk mixture, stirring constantly.