In a 4-quart pot cook potatoes, onion, and garlic in lightly salted boiling water for 15 to 20 minutes or until very tender; drain and return to pot.
Add about half of the chicken broth. Use an immersion blender to puree the potato mixture in the pot. (Or, transfer mixture to a blender; cover and blend until smooth and return to pot.) Stir in remaining broth, milk, salt, pepper, and chopped lovage. Heat through. Serve drizzled with olive oil and sprinkled with additional pepper and lovage leaves.