We give a $2^{\tilde{O}(\sqrt{n}/ε)}$-time algorithm for properly learning monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Our algorithm is robust to adversarial label noise and has a running time nearly matching that of the state-of-the-art improper learning algorithm of Bshouty and Tamon (JACM '96) and an information-theoretic lower bound of Blais et al (RANDOM '15). Prior to this work, no proper learning algorithm with running time smaller than $2^{Ω(n)}$ was known to exist. The core of our proper learner is a \emph{local computation algorithm} for sorting binary labels on a poset. Our algorithm is built on a body of work on distributed greedy graph algorithms; specifically we rely on a recent work of Ghaffari (FOCS'22), which gives an efficient algorithm for computing maximal matchings in a graph in the LCA model of Rubinfeld et al and Alon et al (ICS'11, SODA'12). The applications of our local sorting algorithm extend beyond learning on the Boolean cube: we also give a tolerant tester for Boolean functions over general posets that distinguishes functions that are $ε/3$-close to monotone from those that are $ε$-far. Previous tolerant testers for the Boolean cube only distinguished between $ε/Ω(\sqrt{n})$-close and $ε$-far.