Heat oil in a medium pot over medium heat. Cook leeks, stirring often until softened, about 5 minutes. Measure out 1/4 cup leeks and set aside. Add garlic to remaining leeks in pot and cook until fragrant, about 2 minutes. Increase heat to high. Add broth and potatoes and cook, covered, until potatoes are tender, 10 to 15 minutes. Add half-and-half and salt.
Working in batches, whirl soup in a blender to purée. Divide soup among 4 bowls and top each with about 1 tbsp. reserved leeks.
Note: Nutritional analysis is per 2-cup serving.