Place eggs in a saucepan and cover with cold water by about 1 inch. Bring to a boil over medium-high heat, then remove from heat, cover, and let stand 10–12 minutes, until yolks are set and whites are firm.
Fill a bowl with ice and water. Transfer them to the ice bath and cool 5–10 minutes, until cold.
Tap each on the counter to crack, peel under running water, pat dry, and coarsely chop into roughly 0.25-inch pieces.
In a medium bowl, whisk mayonnaise, Dijon mustard, lemon juice, kosher salt, and black pepper until smooth.
Stir in celery and chives.
Fold in the chopped pieces until evenly coated and creamy.
Lay out sandwich bread. Arrange butter lettuce on 4 slices.
Spoon the salad onto the lettuce, top with remaining bread, press gently, cut in half, and serve.