Step 1Combine the shallots, lemon juice, vinegar, oil and salt in a small, lidded container and shake well to mix. Taste and add more lemon juice if necessary; the mixture should be very bright. Set aside.
Step 1Trim the dry bases from the romaine and quarter each head lengthwise. Slice the radishes as thinly as possible. Separate the white and yolk of the eggs; finely chop the white and press the yolk through a fine mesh strainer into a bowl.
Step 2Combine the bread crumbs in a small saucepan with the olive oil. Stir to coat well; there should be only a light trace of oil in the bottom of the pan. Season with a pinch of salt and place over medium-high heat. Cook, stirring constantly, until the crumbs have darkened and toasted, about 5 minutes. Transfer them to a small bowl and set them aside to cool slightly.
Step 3When almost ready to serve, add 1 tablespoon chopped egg white to the dressing and shake well to mix again.
Step 4Grill the romaine on a stove-top grill pan over high heat, cooking just long enough to sear, 1 to 2 minutes to a side.
Step 5As each batch comes off the grill pan, arrange it on a platter, season lightly with salt and spoon a generous tablespoon or so of dressing over the top. Repeat until all the romaine has been cooked and added to the platter.
Step 6Distribute the thinly sliced radishes over the top. Scatter over the egg whites and then the egg yolks. Spoon over a little more of the salad dressing, scatter over the toasted bread crumbs and serve.