Preheat oven to 350 degrees Fahrenheit. Peel garlic cloves and put all but 5 cloves onto a rimmed baking sheet and drizzle with olive oil. Bake until fragrant and slightly browned, about 10-12 minutes.

Peel, chop, and boil potatoes until tender. Drain completely and set aside.

In a large pot, melt butter. Mince the remaining 5 cloves of garlic and add that to the pot with the onion. Cook until translucent.

Add potatoes, milk, and broth to the pot. Either using an immersion blender or working in batches with a high-speed blender, thoroughly combine.

Return pureed soup to pot. Stir in cheddar cheese, black pepper and salt.

Taste, adjust seasonings and serve! Top with additional cheddar cheese or snipped chives if desired.