Slice the top third off the head of garlic. Place on a piece of aluminum foil and drizzle with the olive oil. Sprinkle with the thyme leaves and some kosher salt and pepper, then place the top back on the garlic. Wrap in the foil and roast until the garlic is brown and very soft, about 45 minutes.
Meanwhile, cover the potatoes with cold water in a large pot. Add 1 tablespoon of kosher salt to the water and bring to a simmer over medium-high heat, about 15 minutes. Simmer until the potatoes are fork-tender, another 15 to 20 minutes.
Drain the potatoes and return them to the pot. Add the hot chicken broth. Squeeze the roasted garlic out of its skin into a small bowl, drizzling in any olive oil from the foil. Mash the garlic smooth with a fork. Add the garlic to the potatoes and mash until smooth using an immersion blender fitted with the whisk attachment. Add more broth as needed. Once smooth, add three-quarters of the chives and some sea salt and black pepper, then mix through.
Tip into a serving bowl and garnish with the remaining chives.