Neutral-atom quantum computers can now arrange atoms in three-dimensional tweezer arrays, yet every existing compiler assumes a flat geometry. We present Piqasso, a compiler that exploits the vertical axis by stacking storage, entanglement, and readout into distinct layers. Its pipeline pairs an analytical placement respecting axial-clearance optics with a router that brings gate partners together via short vertical hops---bypassing in-plane crossing conflicts through out-of-plane detours---and a multi-AOD scheduler that parallelizes transport across focal planes. On 34 circuits, Piqasso reduces atom transport distance by 2.1$\times$ over a state-of-the-art planar compiler, yielding up to 7.3$\times$ faster execution, 2.2$\times$ higher movement fidelity, and 1.8$\times$ fewer serialized transport rounds, with all gains widening at scale.

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