In a medium bowl, combine the shallot, lemon juice, sugar, and 1/2 tsp. salt. Stir to dissolve the sugar and salt and then let the mixture sit for 15 min.
Meanwhile, cut off the root end of the romaine so that the leaves fall loose. Wash the leaves, spin them dry, and put them in a large bowl.
Slowly whisk the olive oil into the lemon juice mixture (you should have about 1/2 cup vinaigrette). Season with salt, if necessary, and a few grinds of pepper. (This vinaigrette can be made ahead; just whisk to recombine it before using. If you make it more than 2 hours ahead, cover, refrigerate, and bring it to room temperature before using.)
Toss the romaine with enough vinaigrette to lightly coat. Season the salad with salt and pepper to taste. Arrange the leaves on a large platter and sprinkle with the shaved Parmigiano.