Prick the potatoes several times all over with a fork.
Cook the potatoes over high heat for five 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 200c oven for about an hour or until tender. Set the cooked potatoes aside and cool to room temperature.
While the potatoes are cooling, make the spread: In a bowl, whisk together the American Garden mayonnaise, parmesan cheese, garlic and rosemary. Season with half teaspoon salt and several grinds of pepper, or to taste.
Cut the cooled potatoes crosswise into two-centimeter thick slices. Liberally brush the spread on each cut side of the potato slices.
Heat a grill over medium-high heat until hot. Place the potato slices on the grill racks. Grill the potatoes for one to two 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.
Remove the potatoes to a plate and cool slightly before serving.