Cut about 1/2 inch off the top of the heads of garlic, exposing the individual cloves. Drizzle the tops with olive oil, wrap in foil, and place in the oven for 35-40 minutes, until soft. Remove from the oven and allow to cool slightly; once cool enough to handle, squeeze the softened cloves into a bowl.
Meanwhile, peel the potatoes and cut into similarly-sized chunks. Place into a large pot, cover with cold water, salt generously, and bring to a boil. Cook for 20-25 minutes, until a paring knife inserted into the middle of the potatoes goes in easily. Drain and return to the hot pot, allowing any extra water to evaporate from the potatoes.
Add the butter, roasted garlic, and about 1/2 cup of the buttermilk and mash the potatoes, adding the extra butter as needed until you reach your desired consistency. Stir in the crumbled goat cheese and chives; season with salt and pepper to taste. Serve warm.