We introduce and study a model of directed last-passage percolation in planar layered environment. This environment is represented by an array of random exponential clocks arranged in blocks, for each block the average waiting times depend only on the local coordinates within the block. The last-passage time, the total time needed to travel from the source to the sink located in a given block, maximized over all the admissible paths, becomes a stochastic process indexed by the number of blocks in the array. We show that this model is integrable, particularly the probability law of the last-passage time process can be determined via a Fredholm determinant of the kernel that also appears in the study of products of random matrices. Further, we identify the scaling limit of the last-passage time process, as the sizes of the blocks become infinitely large and the average waiting times become infinitely small. Finite-dimensional convergence to the continuous-time critical stochastic process of random matrix theory is established.