Make aioli (DO AHEAD, can be made up to 2 days ahead): Whisk mayonnaise, oil, oregano, garlic, lemon juice, and lemon peel in small bowl to blend. Season aioli to taste with salt and freshly ground black pepper. Cover and chill.
Prep artichokes: Bring large saucepan of salted water to boil. Fill large bowl with cold water; squeeze juice from lemon halves into water. Working with 1 artichoke at a time, pull off all dark green outer leaves until only tender inner yellow leaves remain. Cut off top 1 inch from artichoke. Cut off stem and trim around stem base of artichoke. Cut artichoke lengthwise in half and immediately drop into lemon water to prevent discoloration. Repeat with remaining artichokes.
Blanch artichokes: Drain artichokes, then add to saucepan of boiling salted water and blanch 3 minutes. Drain again. Cover and chill.
Preheat oven to 425°F. Line 2 rimmed baking sheets with foil. Place artichokes on 1 prepared baking sheet. Drizzle 1 tablespoon oil over; sprinkle with salt and pepper and toss to coat. Spread in even layer on sheet. Place asparagus spears on second baking sheet. Drizzle remaining oil over; sprinkle with salt and pepper and toss to coat. Spread in even layer on sheet.
Roast asparagus and artichokes until tender, about 8 minutes for asparagus and about 15 minutes for artichokes. Arrange artichokes and asparagus on platter. Place bowl of aioli in center for dipping and serve.