Carefully rinse the romaine hearts and pat them dry with paper towels. Trim off just the dark edge of each root end, making sure to leave the romaine hearts intact.
Roughly chop your walnuts and place them in a small pan on the stove over medium-high heat. Stir them around for two minutes, just until they become fragrant. Remove the pan from the heat.
Brush your grill pan with olive oil and heat it (or your outdoor grill) on medium high. When the grill is hot, add the romaine hearts and cook them for three minutes before flipping them over. Cook the romaine for three minutes longer, turning if needed to get it browned all around.
While the romaine is cooking, use a vegetable peeler to shave off pieces of Parmesan and set them aside.
To make your vinaigrette, whisk the olive oil, vinegar and seasonings in a small container. Serve the romaine topped with the walnuts, Parmesan and some dressing.
I've found it's easier to buy a package of romaine hearts, because they are cleaner than a head of romaine. If you are starting with a head of romaine, remove the outer leaves and any wilted leaves until you find the hearts in the middle.