Make the mint oil
In a small saucepan over medium, heat the oil. Break up the mint sprigs and add to the saucepan. Heat until mint is fragrant, 10 to 15 minutes. Remove from heat and strain the mint leaves. Transfer oil to a squeeze bottle.
Make the soup
In a large skillet over medium heat, melt the butter. When butter is bubbling, add leeks, garlic and 1/4 teaspoon salt. Cook until leeks and garlic have softened, 10 to 12 minutes. Add remaining soup ingredients except lemon juice to the blender in the order listed. Add leek and garlic mixture. Select the SOUP setting and press start. Once the cycle finishes, add lemon juice and PULSE several times to combine.