Peel the potatoes, and cut into large pieces.
Heat the chicken broth and potatoes to boiling in the saucepan over high heat, stirring occasionally with a fork to make sure potatoes do not stick to the saucepan. Once mixture is boiling, reduce heat just enough so mixture bubbles gently. Cover and cook about 15 minutes or until potatoes are tender when pierced with a fork.
While the potatoes are cooking, peel and thinly slice the green onions. If you have extra onions, wrap them airtight and store in the refrigerator up to 5 days.
When the potatoes are done, remove the saucepan from the heat, but do not drain. Break the potatoes into smaller pieces with the potato masher or large fork. The mixture should still be lumpy.
Stir the milk, salt, pepper, thyme and onions into the potato mixture. Heat over medium heat, stirring occasionally, until hot and steaming, but do not let the soup boil.