Preheat your oven to 375–400 °F (190–200 °C). Slice the top off the head of garlic, drizzle with olive oil, and season with salt and pepper. Wrap it in foil and roast for 40–45 minutes.

In a large pot, bring water to a boil. Add the Russet potatoes and cook until fork-tender, about 15–20 minutes. Drain and let cool slightly.

In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onions and leeks, sautéing for 5–7 minutes until soft and translucent.

Lower the heat and stir in the flour, cooking for about 1 minute. Gradually whisk in the broth, milk, cream, and thyme, seasoning with salt and pepper. Heat until the mixture thickens slightly.

Add the cooked potatoes and roasted garlic cloves. Use an immersion blender to purée the soup until smooth. Adjust consistency with broth as needed and season to taste.

Butter one side of each bread slice. Layer Gruyère cheese on the non-buttered side of two slices, then top with remaining bread slices (buttered side out). Cook in a skillet over medium heat for about 5 minutes per side.

Ladle the warm soup into bowls, garnish with Parmesan cheese and chives, and top with grilled cheese croutons. Serve immediately.