Final answer:
The radial velocity method using Doppler shifts is best for detecting an exoplanet several thousand light-years away, especially if it is a large planet close to its host star. Direct imaging is ineffective due to faintness and glare, while the transit method might not be applicable depending on the orientation of the planet's orbit.
Step-by-step explanation:
The most suitable exoplanet detection method for finding a planet several thousand light-years away with a semi-major axis of a few tens of AU and a mass of several Earth masses is the radial velocity, or Doppler effect technique. This method measures the variations in the star's radial velocity caused by the gravitational pull of an orbiting planet, leading to a detectable wobble in the star's motion. It is particularly sensitive to large planets in close orbits, which create significant gravitational effects on their host stars.
Direct imaging of such planets is generally ineffective due to the planets being faint and lost in the star's glare. Moreover, the transit method employed by telescopes like Kepler may not be suitable if the distant star's brightness variations are too small to detect or if the planet does not transit from our viewpoint. However, if the planet's mass is substantial enough to cause a measurable Doppler shift, ground-based telescopes can detect it, as has been done for hundreds of other planets.