Preheat oven to 400 degrees, with racks in center and lower thirds. Put garlic on a small piece of foil; drizzle with oil. Fold to form a packet, and crimp edges to seal. Place garlic packet and the potatoes on center rack; place a baking sheet on lower rack to catch the juices.
Bake, flipping halfway through, until the sweet potatoes are very tender when pierced with the tip of a knife, about 45 minutes. Remove sweet potatoes and garlic from oven; set aside. Continue to bake russet potatoes until very tender, about 15 minutes more. Remove from oven; let cool slightly, about 5 minutes.
Peel potatoes. Pass through a ricer or a food mill into a medium bowl. Squeeze garlic from skins into bowl with potatoes. Stir in butter, parsley, sage, salt, and pepper. Stir in yogurt, and serve.