In a medium saucepan over medium-low heat, add the half-and-half, garlic, thyme, and butter and heat until the butter is melted. Allow to sit over very low heat to infuse the flavors and keep hot.
In a large pot, add the potatoes, cover with water and generously season with salt. Bring to a boil over medium heat, and then reduce the heat and simmer until the potatoes are fork tender, about 15 to 20 minutes. Immediately drain the potatoes and put them through a food mill over a serving bowl. If you don't have a food mill, you can hand mash or use a hand mixer, but be sure to not over-whip if using an electric hand mixer or the potatoes will get gummy.
Pour the hot infused half-and-half mixture through a fine mesh sieve into the potatoes. Stir with a wooden spoon to combine and season with a little salt and pepper. Taste and adjust seasoning, if needed. Eat and enjoy!
Serve the potatoes in a warm serving vessel to help keep them hot.
Keep the butter wrapper to cover the top of the whipped potatoes, butter side down, to keep them moist if they are not being served immediately.