Metal contacts to two-dimensional (2D) semiconductors are ubiquitous in modern electronic and optoelectronic devices. Such contacts are, however, often plagued by strong Fermi level pinning (FLP) effect which reduces the tunability of the Schottky barrier height (SBH) and degrades the performance of 2D-semiconductor-based devices. In this work, we show that monolayer MoSi$_2$N$_4$ and WSi$_2$N$_4$ - a recently synthesized 2D material class with exceptional mechanical and electronic properties - exhibit strongly suppressed FLP and wide-range tunable SBH when contacted by metals. An exceptionally large SBH slope parameter of S=0.7 is obtained, which outperform the vast majority of other 2D semiconductors. Such surprising behavior arises from the unique morphology of MoSi$_2$N$_4$ and WSi$_2$N$_4$. The outlying Si-N layer forms a native atomic layer that protects the semiconducting inner-core from the perturbance of metal contacts, thus suppressing the FLP. Our findings reveal the potential of MoSi$_2$N$_4$ and WSi$_2$N$_4$ monolayers as a novel 2D material platform for designing high-performance and energy-efficient 2D nanodevices.