Cook potatoes and garlic in large pot of boiling water until potatoes are very tender, about 15 minutes. Drain; return potatoes and garlic to same pot.
Meanwhile, melt butter in heavy small saucepan over medium heat. Add chopped prosciutto and thyme and sauté until fragrant, about 2 minutes.
Add prosciutto mixture and 3/4 cup milk to potatoes and garlic. Mash well, adding more milk by tablespoonfuls if potatoes are dry. Mix in 1/2 cup cheese. Season with pepper. (Can be prepared 6 hours ahead. Cover and chill. Stir over low heat to rewarm, adding more milk by tablespoonfuls, if desired.) Transfer potatoes to bowl. Sprinkle lightly with 2 tablespoons cheese; serve.