In a roomy saute pan with a tight-fitting lid, bring carrots, ¼ teaspoon salt, and water to a boil. Reduce heat to medium and cover tightly. Simmer 6 to 8 minutes, until just tender, then transfer carrots to a bowl and set aside to cool.
Put the leeks on metal skewers, drizzle with 1 tablespoon olive oil, sprinkle with ¼ teaspoon salt, and place on a charcoal grill or under the broiler. Cook, turning frequently, about 10 minutes, or until the outer skin is lightly charred in patches. Set aside; when cool to the touch, remove from skewers and slice each bulb in half lengthwise. Set aside.
Next, make the dressing; set aside.
When ready to serve, divide the prosciutto among six plates and drizzle with the remaining olive oil. Drain excess salad dressing from the carrots and drizzle over spinach leaves. Place ½ cup spinach leaves in the center of each plate; over them, arrange 6 leek halves in a star pattern, then pile carrots in the middle. Sprinkle with basil leaves. Serve at room temperature.