Prepare barbecue to medium-high heat. Mix together 2 tablespoons olive oil and 1 tablespoon fresh lemon juice. Brush artichokes with olive oil mixture and grill until tender and lightly charred, turning occasionally, about 10-15 minutes.
To prepare aioli, mix together mayonnaise, lemon zest, 1 tablespoon fresh squeezed lemon juice, olive oil, garlic, salt, white pepper, paprika and cayenne in small bowl.
Serve artichokes warm or at room temperature with lemon garlic aioli.