Gently boil egg-cooking water in a saucepan. Carefully lower in eggs; simmer 11 minutes. Chill in ice and cold bath water 10 minutes.
Peel eggs; separate yolks and whites. Press yolks through a fine sieve; mix with mayonnaise, milk, salt, and white pepper until smooth.
Finely chop whites; fold into yolk mixture. Cover; refrigerate 30 minutes until thoroughly cold, included in prep time.
Thinly butter one face of each bread slice. Divide salad over 2 slices; smooth evenly almost to edges.
Close with remaining bread, butter-side down. Neatly trim crusts; cut diagonally. Serve cold or promptly refrigerate at 40°F or below.