In medium saucepan, combine potatoes, parsnips, and thyme sprigs. Cover with cold water and bring to a boil. Add 2 teaspoons salt, reduce heat, and simmer until tender, 12 to 15 minutes. Reserve 1/2 cup cooking liquid and drain; discard thyme sprigs. Return vegetables to pot. Mash with butter, adding 2 tablespoons reserved cooking liquid at a time, until smooth.

While vegetables cook, heat broiler. Heat oil in cast-iron skillet on medium. Add onion and cook, covered, stirring occasionally, 3 minutes.

Add carrots and cook, covered, until just tender, 5 to 6 minutes. Stir in garlic, then add lamb and cook, breaking up with spoon into small pieces, until no longer pink, 3 to 5 minutes.

Stir in tomato paste, cinnamon, and 1/2 teaspoon salt and pepper; cook 1 minute. Stir in beer and water, then bouillon base; simmer until thickened, about 3 minutes. Fold in parsley and thyme leaves.

Spread potatoes over lamb and sprinkle with additional thyme. Broil until golden brown, 2 to 4 minutes.