Trim the asparagus bottoms about 2 inches and peel the outside of the bottoms of the stalks if they are thick asparagus. Place the asparagus in a baking pan. Don’t worry about finding a pan large enough to hold the asparagus in a single layer – the stalks can overlap a bit. Drizzle the olive oil over them and toss gently to coat the asparagus evenly. Sprinkle on the salt, and toss again.
Roast the asparagus for 7 to 10 minutes for thin stalks, 10 to 14 for thick ones. Remember that they will continue to cook a bit after you remove them from the oven, so take them out while they’re still a little firmer than you would like. Also, it’s very hard to dip a super tender asparagus. Transfer to a plate and let cool to room temperature.
While the asparagus are roasting, make the dipping sauce. Place the arugula, scallions, basil, and salt and pepper in a food processor or blender and pulse to chop, then run the machine to chop finely.
In a medium bowl combine the mayonnaise, sour cream, lemon juice and Worcestershire sauce, if using. Turn the herb mixture into the bowl and stir well to combine. Transfer to a small serving bowl and serve with the asparagus for dipping.