Preheat oven to 350 degrees F. Wash, dry, and then slit the potatoes in 1/8 inches apart slits straight down into the potato, but stop just short of cutting all the way through. Try using the long handle of a wooden spoon in front of the potato as a guide on where to stop.
Melt 2 tbsp butter in a cast iron or stainless steel skillet, mince a clove of garlic and scrape into the pan and toss in a sprig of rosemary. Brown for just a minute, then put the potatoes in the skillet and baste them with the garlic butter sauce. Put skillet in the oven and bake for 45 minutes.
Remove skillet from oven and turn the oven up to 450 degrees. Add 2 tbsp of butter, another clove of minced garlic, and another sprig of rosemary and baste the potatoes again. When the oven is preheated, put skillet back in the oven for another 15 minutes
Take out the skillet and throw in some cherry tomatoes and thyme. Baste and salt and pepper everything again.
Bake for another 15 minutes. Serve with the butter and crumbly bits from the pan and fresh parmigiano on top.