Before You Begin
Prepare your cookware as needed. Cooking times may vary depending on your equipment and ingredients.
Preheat oven to 375°F (190°C). Slice the top off the garlic head, drizzle with olive oil, season lightly with salt and pepper, wrap in foil, and roast for 40–45 minutes, until cloves are very soft and golden. Cool slightly, then squeeze out the garlic.
Bring a pot of water to a boil. Add diced potatoes and cook until fork-tender, about 12–15 minutes. Drain and set aside.
In a large pot over medium heat, melt butter. Add onions and leeks and cook 6–7 minutes, until softened and fragrant.
Stir in flour and cook for 1 minute, stirring constantly. Gradually whisk in vegetable broth, milk, cream, thyme, salt, and pepper. Bring to a gentle simmer.
Add cooked potatoes, roasted garlic, and Parmesan. Blend carefully with an immersion blender until smooth. If using a countertop blender, blend in batches and vent the lid to avoid pressure buildup.
Continue heating gently until warmed through and steaming, reaching 165°F (74°C). Do not boil after adding dairy. Adjust thickness with additional warm broth if needed.
For the croutons, butter the bread, add cheese, and grill over medium heat until golden brown and cheese is melted. Cut into cubes.
Ladle soup into bowls and top with grilled cheese croutons and chives. Serve immediately.