In this paper, we propose a broadband-tunable resonant-cavity-enhanced photodetector (RCE-PD) structure with double high-index-contrast grating (HCG) mirrors and numerically investigate its characteristics. The detector is designed to operate at 1550-nm wavelength. The detector structure consists of a top InP HCG mirror, a p-i-n photodiode embedding multiple quantum wells, and a Si HCG mirror formed in the Si layer of a silicon-on-insulator wafer. The detection wavelength can be changed by moving the top InP HCG mirror suspended in the air. High reflectivity and small penetration length of HCGs lead to a narrow absorption linewidth of 0.38 nm and a broad tuning range of 111 nm. The peak absorption efficiency is 76-84% within the tuning range. This broadband-tunable and narrow-absorption-linewidth RCE-PD is desirable for applications where selective wavelength demultiplexing is required. Furthermore, the fact that it can be fabricated on a silicon platform offers us a possibility of integration with electronics.