To give this dish its distinctive flavor, try using Spanish pimenton dulce (sweet pimento).
Fill a large bowl with ice water. Bring a stockpot of water to a boil. Once it boils, salt water generously (it should taste like the ocean). Add asparagus; cook at a gentle boil until tender, 6 to 7 minutes. Transfer to ice bath to stop the cooking. Transfer chilled asparagus to a paper-towel-lined plate.
Meanwhile, whisk lemon juice, egg yolk, pimenton, and 1/2 teaspoon salt in a bowl. Slowly add olive oil a drop at a time, whisking constantly. Once it begins to emulsify, whisk in oil in a slow, steady stream. Let aioli stand at least 30 minutes at room temperature so flavors meld. To serve, drizzle on cooled asparagus.