Roast the Garlic: In a large soup pot, add garlic cloves and enough olive oil to submerge the garlic cloves. Cook covered over medium-low heat, stirring frequently, until fork tender around 25-45 minutes. Be careful not to burn them! If garlic sticks, add oil, water, or white wine and scrape the bottom of the pot.
Sauté Onions: In the same pot, add butter, onions, salt and pepper, and cook covered over medium-high heat until onions turn translucent, around 5-7 more minutes. Use white wine as needed for stuck onions and scrape the bottom of the pan.
Build the Broth: Add chicken or vegetable broth, potatoes, thyme, and bay leaves. Bring the pot to a boil over high heat, then reduce to low and simmer, covered, until potatoes are fork tender, around 15-25 minutes.
Blend with Final Touch of Cream: Remove bay leaves, and remove pot from heat and let cool for 5-10 minutes. Add heavy cream. Use an immersion blender and blend until smooth, or transfer to a blender in batches. Stir, serve with parmesan cheese, and enjoy!
