Step 1Place the 12 eggs in a large saucepan. Fill with cool water. Bring the water to boil over medium heat. As soon as it boils, reduce the heat to simmer and cook the eggs 20 minutes. Fill a bowl with ice cubes and cold water and set aside.
Step 2Meanwhile, place the sun-dried tomatoes in a microwave-safe bowl and cover them with water. Microwave for 2 minutes, then let the tomatoes sit several minutes until they have softened. Drain on paper towels, then finely mince.
Step 3When the eggs are cooked, lift them from the pan using a slotted spoon and place them in the cold water. Chill completely, then peel and slice them in half lengthwise. Remove the yolks and push them through a potato ricer or a mesh strainer to break them up into small pieces. This will help make a smoother mixture.
Step 4Combine the yolks, tomatoes, capers, shallot, mustard, mayonnaise and 2 tablespoons of basil in a small bowl. Stir to combine.
Step 5If the egg whites are inclined to roll around on a plate, cut a very thin slice off the bottom. Fill a pastry bag fitted with the star tip with the egg yolk mixture. Pipe the filling into the egg whites, then garnish the eggs with the remaining basil. If you don't have a pastry bag, you can cut the tip off a resealable bag to pipe in the filling, or use a teaspoon. Cover the eggs with plastic wrap and refrigerate until ready to serve.