TO MAKE SOUP BASE: Melt butter in Dutch oven over medium heat. When foaming subsides, add leeks and cook until soft (do not brown), 5 to 8 minutes. Stir in minced garlic and cook until fragrant, about 1 minute. Add garlic heads, 6 cups broth, bay leaves, and 3/4 teaspoon salt; partially cover pot and bring to simmer over medium-high heat. Reduce heat and simmer until garlic is very tender when pierced with tip of knife, 30 to 40 minutes. Add potatoes, partially cover pot, and bring mixture to simmer over medium-high heat. Take pot off heat and let rest 10 minutes. Separate soup into 2 storage containers and freeze up to 1 month.
TO REHEAT AND SERVE: Run hot water over surface of storage containers to help release frozen blocks of soup. Add soup and 1 cup broth to large Dutch oven set over medium-high heat. Cover and cook, stirring occasionally, until soup is hot and potatoes are tender, about 20 minutes. Discard bay leaves. Remove garlic heads; using tongs or paper towels, squeeze garlic heads at root end until cloves slip out of their skins. Using fork, mash garlic to smooth paste in bowl.
3. Stir cream, thyme, and half of mashed garlic into soup; heat soup until hot, about 2 minutes. Taste soup; add remaining garlic paste if desired. Using immersion blender, process soup until creamy, with some potato chunks remaining. Alternatively, transfer 1 1/2 cups potatoes and 1 cup broth to blender or food processor and process until smooth. (Process more potatoes for thicker consistency.) Return puree to pot and stir to combine. Season with salt and pepper and serve, sprinkling each portion with chives and garlic chips.
For Garlic Chips: Heat oil and garlic in 10-inch skillet over medium-high heat. Cook, turning frequently, until light golden brown, about 3 minutes. Using slotted spoon, transfer garlic to plate lined with paper towels; discard oil. Sprinkle lightly with salt.