State-of-the-art hypergraph partitioners follow the multilevel paradigm that constructs multiple levels of progressively coarser hypergraphs that are used to drive cut refinements on each level of the hierarchy. Multilevel partitioners are subject to two limitations: (i) Hypergraph coarsening processes rely on local neighborhood structure without fully considering the global structure of the hypergraph. (ii) Refinement heuristics can stagnate on local minima. In this paper, we describe SpecPart, the first supervised spectral framework that directly tackles these two limitations. SpecPart solves a generalized eigenvalue problem that captures the balanced partitioning objective and global hypergraph structure in a low-dimensional vertex embedding while leveraging initial high-quality solutions from multilevel partitioners as hints. SpecPart further constructs a family of trees from the vertex embedding and partitions them with a tree-sweeping algorithm. Then, a novel overlay of multiple tree-based partitioning solutions, followed by lifting to a coarsened hypergraph, where an ILP partitioning instance is solved to alleviate local stagnation. We have validated SpecPart on multiple sets of benchmarks. Experimental results show that for some benchmarks, our SpecPart can substantially improve the cutsize by more than 50% with respect to the best published solutions obtained with leading partitioners hMETIS and KaHyPar.