Slice part of one zucchini into julienne strips to equal ½ cup strips, cutting each julienne strip about 1 to ½ inches long. Set julienne strips aside to use as a garnish.
Quarter all the remaining zucchini lengthwise, and then cut into ½ to 1 inch cubes. Set zucchini cubes aside.
Heat olive oil in a Dutch oven over medium high heat. Add onion and carrot, and cook, stirring frequently about 3 to 5 minutes or until onion is tender. Add jalapeño and garlic, cook 1 minute.
Add broth and zucchini cubes. Season lightly with salt and generously with pepper. Heat until boiling. Reduce heat, cover and simmer 30 minutes or until zucchini is very tender.
Add basil and cook 2 to 3 minutes.
Remove soup from heat. Carefully, using an immersion blender, blend soup until smooth. Reheat soup if necessary.
To serve, ladle into serving bowls. Top each serving with julienne strips of zucchini and sprinkle with 1 tablespoon fresh grated Parmesan cheese.