Preheat the oven to 350 degrees F. To roast the garlic, slice the tops of the heads of the bulbs off. Use your fingers to rub the “paper” off of the sides – as much as you can get off is fine. Drizzle the exposed cloves with olive oil and wrap each bulb in aluminum foil. Roast for 45 to 50 minutes, or until the exposed cloves are golden and caramely. Remove and let cool until you can safely touch them. Once cool, squeeze the roasted cloves out of the bulbs, making sure to remove any paper.
Add the potatoes to a large pot and fill it with cold water – enough to cover an inch or two above the potatoes. Bring to a boil and boil until the potatoes are fork-tender, about 25 minutes or so (depending on the size of your potato chunks). Once tender, drain the potatoes.
Add the potatoes to the bowl of your electric mixer (you can obviously use a hand masher or a hand mixer if desired) fitted with the paddle attachment. Mix for a minute or two until the potatoes start to break down and come together, then add in the goat cheese (I like to leave a chunk out for topping) and garlic cloves. Mix on medium speed until the potatoes are creamy. Reduce the speed to low and mix in the half and half, butter, salt and pepper. Mix for another few minutes until totally creamy and no chunks remain. Taste and season additionally if desired – you may need more salt and pepper!
Spoon the potatoes into a bowl and top with the chives and extra piece of goat cheese. Serve immediately.