How to Make It
In a large saucepan, cover the eggs with cold water and bring to a rapid boil over moderately high heat. Cover and remove the saucepan from the heat; let stand for 12 minutes.
Meanwhile, in a small bowl, combine the minced shallot with the vinegar and let soak for 10 minutes. Drain well and discard the vinegar.
Drain and rinse the eggs under cold water, shaking the pan vigorously to crack the shells. Let the eggs cool completely. Peel the eggs and pat dry, then coarsely chop.
In a large bowl, combine the mayonnaise, mustard, celery, capers, chives, lemon juice and vinegar-infused minced shallot. Add the eggs and mix gently with a wooden spoon. Season the egg salad with salt and pepper and refrigerate.
Heat the oil in a large skillet. Add the sliced shallot and cook over high heat, stirring, until golden, 2 to 3 minutes. Add the spinach and cook, stirring until wilted, 2 to 3 minutes. Drain the spinach and press out the water. Coarsely chop the spinach and transfer it to a bowl. Season with salt and pepper and refrigerate until chilled.
Arrange the toasted brioche slices on a work surface. Spoon the egg salad on half of the slices and top with the spinach. Close the sandwiches and cut in half. Serve right away.