We theoretically demonstrate the nontrivial transmission properties of a graphene-insulator-metal waveguide segment of deeply subwavelength scale. We show that, at midinfrared frequencies, the graphene-covered segment allows for the resonant transmission through the graphene-plasmon modes as well as the nonresonant transmission through background modes, and that these two pathways can lead to a strong Fano interference effect. The Fano interference enables a strong modulation of the overall optical transmission with a very small change in graphene Fermi level. By engineering the waveguide junction, it is possible that the two transmission pathways perfectly cancel each other out, resulting in a zero transmittance. We theoretically demonstrate the transmission modulation from 0% to 25% at 7.5-mu m wavelength by shifting the Fermi level of graphene by a mere 15 meV. In addition, the active region of the device is more than 50 times shorter than the free-space wavelength. Thus, the reported phenomenon is of great advantage to the development of on-chip plasmonic devices.