Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix $S_{ij}=\mathrm{\text{Tr}}(O_i\rho (t_j))$ for a collection of $M$ low-weight Pauli observables $\{O_i{\}}_{i=1}^M$ over $N$ timesteps $\{t_j{\}}_{j=1}^N$, while minimizing the total number of device shots. We propose a (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples $m\ll N$ timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-$N$ signal via standard ${\ell }_1$-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly $s$-sparse signals in a unitary transform basis, we show that $m=𝒪(s{log}^{2}slogN)$ random timesteps suffice (with high probability), leading to total-shot savings scaling as $\tilde{\Theta }(N/s)$ (i.e., up to polylogarithmic factors) relative to collecting shadows at all $N$ timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.