We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has the second eigenvalue bounded away from 1. Consequently, we obtain a polynomial time algorithm to find linear-sized independent sets in one-sided expanders that are almost -colorable or are promised to contain an independent set of size . Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. Somewhat surprisingly, we observe that the analogous task of finding a linear-sized independent set in almost -colorable one-sided expanders (even when the second eigenvalue is ) is NP-hard, assuming the Unique Games Conjecture. All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques (following the classical spectral methods of Alon and Kahale) and require two-sided expansion, meaning a bounded number of negative eigenvalues of magnitude . Such techniques naturally extend to almost -colorable graphs for any constant , in contrast to analogous guarantees on one-sided expanders, which are Unique Games-hard to achieve for . Our rounding builds on the method of simulating multiple samples from a pseudodistribution introduced by Barak et. al. for rounding Unique Games instances. The key to our analysis is a new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system.