Final answer:
The transit method has detected the most extrasolar planets, outperforming other techniques such as the Doppler method, with missions like NASA Kepler playing a significant role in these discoveries.
Step-by-step explanation:
The method that has detected the most extrasolar planets, or exoplanets, is the transit method. This technique involves measuring the slight dimming of a star as a planet passes in front of it, or transits from our perspective. While the Doppler method, which detects the gravitational tug a planet exerts on its star, was the first to yield many detections, the transit method, through missions like the NASA Kepler space observatory, has found a larger number of candidates due to its ability to continuously monitor tens of thousands of stars. The Kepler mission was specifically designed to discover exoplanets by using this technique and successfully found many by detecting repeated transits, indicating planets with regular orbital periods.