We carry out 3D hydrodynamical simulations to study planet-disc interactions for inclined high-mass planets, focusing on the disc's secular evolution induced by the planet. We find that, when the planet is massive enough and the induced gap is deep enough, the disc inside the planet's orbit breaks from the outer disc. The inner and outer discs precess around the system's total angular momentum vector independently at different precession rates, which causes significant disc misalignment. We derive the analytical formulae, which are also verified numerically, for: (1) the relationship between the planet mass and the depth/width of the induced gap, (2) the migration and inclination damping rates for massive inclined planets, and (3) the condition under which the inner and outer discs can break and undergo differential precession. Then, we carry out Monte Carlo radiative transfer calculations for the simulated broken discs. Both disc shadowing in near-infrared images and gas kinematics probed by molecular lines [e.g. from the Atacama Large Millimeter/submillimeter Array (ALMA)] can reveal the misaligned inner disc. The relationship between the rotation rate of the disc shadow and the precession rate of the inner disc is also provided. Using our disc breaking condition, we conclude that the disc shadowing due to misaligned discs should be accompanied by deep gaseous gaps (e.g. in Pre/Transitional discs). This scenario naturally explains both the disc shadowing and deep gaps in several systems (e.g. HD 100453, DoAr 44, AA Tau, and HD 143006) and these systems should be the prime targets for searching young massive planets (>M-J) in discs.
