You can also serve this soup cold or at room temperature. Puree it in batches to prevent overflow (particularly because you do so while it's hot), if needed. You can stir in a small amount of water to thin the soup after pureeing it, if you like.
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.