Put potatoes in a large pot and fill pot with water until water just covers the potatoes. Bring the water to boil and boil potatoes until they can easily be pierced with a fork, about 20 minutes.
While the potatoes are cooking, in a small saucepan combine the half and half and rosemary. Bring to a simmer. Keep warm until potatoes are done.
When the potatoes are finished, drain and put back in the same pot . Mash until smooth, then add the half and half and rosemary. Stir to incorporate. Then add the butter and stir to incorporate the butter.
Season to taste with salt and pepper.