Heat 4 tablespoons of extra-virgin olive oil in a heavy large pot over medium-low heat. Add leeks, potatoes, and onion. Saute until onion is tender, stirring occasionally, about 12 minutes. Add broth and bring soup to a boil. Reduce heat to medium-low. Simmer until all vegetables are tender, about 20 minutes. Working in batches puree cooled soup in a blender ( or use a hand emulsifier in the pot). Return puree to pot. Season with salt and pepper. (Can be made 1 day ahead. Refrigerate uncovered until cold. Cover and keep refrigerated).
GARLIC CHEESE TOASTS:
Preheat oven to 350°F.
Stir olive oil and garlic in a small skillet over low heat until garlic is fragrant, about 1 minute; remove from heat and cool. Add cheese to garlic in skillet; toss and combine. Arrange bread slices on baking sheet. Spoon cheese mixture onto bread slices, dividing equally. Bake toasts until cheese melts, about 10 minutes.
Bring soup to a simmer over medium heat, stirring frequently. Ladle into bowls. Sprinkle generously with chives. Serve with the garlic cheese toasts.