Mince chorizo. Cook with oil in a small frying pan over medium heat, stirring often, until crisp, 3 to 5 minutes. Strain into a bowl; reserve oil and chorizo separately.
Halve eggs crosswise (not lengthwise) and scoop yolks into a food processor. Pulse 5 seconds. Add 1 tbsp. water, the allioli, lemon zest and juice, espelette chile, smoked paprika, and 2 tsp. reserved chorizo oil (you will probably use it all). Pulse 20 seconds, or until well combined. Season with 1/2 tsp. salt, transfer to a resealable plastic bag, and squish into one corner of bag.
Sprinkle inside of egg-white halves lightly with salt. Snip tip off corner of bag of filling to make a 1/4-in.-wide hole. Pipe about 1 1/2 tbsp. filling into each half. Top with chorizo and chives.
*Find fruity, mild espelette chile on spanishtable.com and smoked sweet Spanish paprika in many grocery stores (often labeled "smoked paprika") and on spanishtable.com.
Make ahead: Whites and filling, up to 1 day, chilled separately. Let whites come to room temperature before filling and serving.