Remove woody ends from asparagus. Peel stalks thinly. Half fill a large saucepan with water, add sugar and salt and bring to a boil. Add asparagus, cook 18 to 20 minutes, depending on thickness, or until crisp–tender. Remove asparagus from pan. Drain; keep warm until serving.
Cut ham into wide strips and arrange over base of serving dish.
To make lemon and garlic mayonnaise, pound garlic and salt to a fine paste in a medium bowl. Add egg yolk and 1/2 tablespoon lemon juice. Whisk vigorously until thickened.
Whisk both olive oils into the mixture, drop by drop, until it starts to thicken. Add remainder in a thin, steady stream until well combined. Add yogurt and remaining lemon juice. Season to taste with salt and white pepper.
Place asparagus on top of ham, coat with a little mayonnaise and sprinkle with parsley leaves. Serve remaining mayonnaise in a separate bowl.
White asparagus is thicker than green asparagus. If using green asparagus, halve the cooking time. The mayonnaise uses white pepper, not black, because a dressing for white asparagus ideally is uniformly pale, with no dark speckles visible to spoil the appearance.