In a medium nonstick skillet, heat oil over medium heat. Add onions, garlic, and salt. Cook, stirring occasionally, 5 minutes, or until onions begin to soften. Cover, reduce heat to medium-low, and cook until onions are very tender, about 15 minutes longer.
Add artichoke hearts, paprika, and cayenne; cook, uncovered, until artichokes are heated through, about 5 minutes. Pour eggs over vegetables, cover, and cook over low heat until set, about 20 minutes.
Uncover the pan and place a large plate over the skillet. Carefully invert the pan and release tortilla onto the plate. Slide inverted tortilla back into the skillet and continue cooking until bottom is golden brown, 5 to 7 minutes. Transfer to a serving plate and cut into quarters. Serve warm.