Fill a 4-qt. pot 1 inch deep with water. Add a steamer basket. Bring the water to boiling. Carefully add the eggs to the basket. Adjust the heat to maintain a steady boil. Cover and steam for 16 minutes.

Remove the eggs with a slotted spoon to a bowl of ice water. When the eggs are cool, crack the shells lightly and remove shells starting from wide ends. (You can cover and chill the peeled eggs up to 5 days in the fridge.)

Halve each egg lengthwise and remove the yolks. Place the yolks in a small bowl; mash completely with a fork. Add mayonnaise, mustard, and the 2 tsp. vinegar; mix well. Stir in the sliced green onions.

Spoon or pipe yolk mixture into the egg white halves. Cover and chill until serving time.

For the pickled vegetables, place the carrots and radishes in a small bowl. In a small saucepan combine the 3/4 cup vinegar, 1/4 cup water, and the sugar. Bring to simmering, stirring until sugar dissolves. Pour the liquid over the carrots and radishes; cool at least 1 hour. (Or cover and chill up to 5 days in the fridge.)

Drain the vegetables, reserving the pickling liquid. For the vinaigrette, in a small bowl whisk together olive oil, 1/4 cup of the reserved pickling liquid, and 1/4 tsp. each salt and freshly ground black pepper.

Arrange the Bibb lettuce on a platter. Top with the pickled vegetables, deviled eggs, and capers. Drizzle with vinaigrette. Serve immediately.