Heat oil in a medium saucepan over medium heat. Add onion, garlic, and 3/4 teaspoon salt. Cook, stirring often, until onion is translucent, about 2 minutes. Add zucchini; cook, stirring occasionally, until skins turn bright green, about 3 minutes. Add stock and 1 cup water; bring to a boil. Reduce heat to medium-low. Simmer, partially covered, until zucchini is tender, about 12 minutes. Let cool slightly.
Puree zucchini mixture in a food processor. Return to pan. Stir in 1 tablespoon mint. Season with salt, if desired. Sprinkle with remaining tablespoon mint. Serve immediately. Keep warm on stove over medium-low heat, if necessary.