Step 1Put the eggs in a small saucepan. Add enough cold water to cover by 1 inch. Bring to a boil over medium-high heat, then cover, remove from the heat and let stand for 6½ minutes. Meanwhile, fill a small bowl with ice and water. When the egg timer goes off, immediately drain the eggs and transfer them to the ice water.
Step 2When the eggs are cool enough to handle, peel and cut in half. Place the celery, mayonnaise, mustard, dill, cayenne, and a quarter teaspoon salt in a large bowl. Using your fingers, tear the egg halves into half-inch pieces into the bowl. Using a rubber spatula, gently fold everything together until evenly mixed. Taste and add more salt and cayenne, if desired.
Step 3Divide the egg salad among bread slices with torn lettuce for sandwiches or scoop into lettuce cups. Top with pickle slices and more dill to serve.