Preheat oven to 400°F.
Scrub potatoes well. Using a paring knife, peel potato skin into strips, taking some flesh with the peel. Strips should be about one and 1/2 inches wide. You should have about 24 strips.
Cut potatoes into cubes and set aside. Place peeled strips on a sheet pan, skin side up. Drizzle with one tablespoon oil or cooking spray.
Roast until edges are crisp, about 25 minutes. Remove from oven.
Heat butter in a six-quart stockpot. Sauté onions over medium heat until slightly-browned, stirring occasionally to keep them from burning.
Add broccoli and cubed potatoes, then add onion powder, nutmeg, salt and pepper. Add six cups water and cook over medium heat for 20 minutes or until broccoli and potatoes are tender.
Add milk and puree soup using a hand blender. Stir in one and 3/4 cups cheddar cheese until melted. Taste and add salt or pepper as desired.
Turn potato skins over and sprinkle with remaining shredded cheddar.
Divide soup among eight soup crocks or heat-proof bowls. Top with three to four potato skins and place all the crocks on a sheet pan. Return to oven for five minutes or until cheese melts. Serve immediately.