In this work, we present new simple and optimal algorithms for solving the variational inequality (VI) problem for \(p^{th}\)-order smooth, monotone operators --- a problem that generalizes convex optimization and saddle-point problems. Recent works (Lin and Jordan (2021), Bullins and Lai (2022), Jiang and Mokhtari (2022)) present methods that achieve a rate of \(\tilde{O}(\epsilon^{-2/(p+1)})\) for \(p\geq 1\), extending results by Nemirovski (2004) and Monteiro and Svaiter (2012) for \(p=1,2\). A drawback to these approaches, however, is their reliance on a line search scheme. We provide the first \(p^{th}\)-order method that achieves a rate of \(O(\epsilon^{-2/(p+1)})\). Our method does not rely on a line search routine, thereby improving upon previous rates by a logarithmic factor. Building on the Mirror Prox method of Nemirovski (2004), our algorithm works even in the constrained, non-Euclidean setting. Furthermore, we prove the optimality of our algorithm by constructing matching lower bounds for a slightly restricted setting. These are the first lower bounds for smooth MVIs beyond convex optimization for \(p > 1\). This establishes a separation between solving smooth MVIs and smooth convex optimization, in addition to settling the oracle complexity of solving \(p^{\textrm{th}}\)-order smooth MVIs.