Spread the sunchokes out evenly on a baking sheet. Season generously with salt and pepper and drizzle with 1/4 cup olive oil. Mix them around with your hands so everything gets evenly coated.
Slice off the top part of the head of garlic to expose a small amount of the cloves. Place the head, root-side down, in a piece of aluminum foil. Season the top cut part with salt and pepper and drizzle with 1 tablespoon olive oil. Wrap the garlic in the foil, creating a package, and place it on the baking sheet with the sunchokes. Roast until the sunchokes are brown and slightly tender, 45 to 50 minutes.
While the sunchokes are still hot, smash them with a potato masher or fork. They should be smashed but not completely mashed up.
In a large bowl, combine the goat cheese, chives, a generous pinch of salt and pepper and the remaining 2 tablespoon olive oil. Add the smashed sunchokes. Take the garlic out of the foil. Squeeze the bulb so that the roasted cloves come out of the skins and fall into the bowl with the goat cheese and sunchokes. Stir it all up until the ingredients are blended and the sunchokes well coated. Serve immediately.