Cook potatoes, covered, in a small amount of boiling, salted water for 8 minutes. Add the asparagus pieces and carrot; cook for 4 to 8 minutes more or until potatoes are just tender and asparagus and carrots are crisp-tender. Drain well.
In a large mixing bowl stir together the pesto and mayonnaise or salad dressing. Add the potatoes, asparagus, carrot strips, green onion, olives, and pepper. Stir until well combined. Cover and chill for 4 to 24 hours.
Just before serving, stir in 1 or 2 tablespoons milk to moisten, if necessary. Sprinkle with Parmesan or Romano cheese. Makes 4 servings.