Preheat the grill to medium. Brush the asparagus with oil and season with salt and pepper, to taste. Grill for 2 to 3 minutes on each side, or until just cooked through. Remove from the grill and place on a serving platter.
Place the cooked egg yolks, tarragon, capers, cornichons, salt, and pepper in the work bowl of a food processor and pulse until just combined. With the processor running, slowly add the oil until emulsified. Add enough white wine vinegar to make the sauce thin enough to drizzle. Taste for seasoning and add more salt and pepper, if necessary. Drizzle the sauce over the asparagus and serve immediately.