Step 1Cook the potatoes: Prick the potatoes several times all over with a fork. Place the potatoes on a paper towel in the microwave. Cook the potatoes over high heat for 5 minutes. Prick the potatoes with a knife to see if they are just tender (do not overcook or the potatoes will fall apart on the grill), cooking an additional minute at a time if needed. The potatoes can also be cooked, on a baking sheet, in a 400-degree oven for an hour or so until tender. Set the cooked potatoes aside and cool to room temperature.
Step 2While the potatoes are cooling, make the spread: In a bowl, whisk together the mayonnaise, Parmesan cheese, garlic and rosemary. Season with ½ teaspoon salt and several grinds of pepper, or to taste.
Step 3Cut the cooled potatoes crosswise into 1-inch thick slices. Liberally brush the spread on each cut side of the potato slices.
Step 4Heat a grill over medium-high heat until hot. Place the potato slices on the grill racks. Grill the potatoes for 1 to 2 minutes, then rotate using a pair of tongs to get good grill marks. After another minute or so, flip the slices over. Grill the potatoes on the other side for a few minutes, rotating partway for good grill marks.
Step 5Remove the potatoes to a plate and cool slightly before serving.