The shape that a compliant microchannel assumes under flow within it is instrumental to understanding the physics governing phenomena ranging from rheological measurements to separation of particles and cells. Fabricating an appreciably rigid microchannel with the desired shape a priori is one possibility for controlling the flow passage geometry, but this does not allow real-time tuning of the system. An alternative approach involves active control of the shape using real-time external stimuli. However, this approach is not viable in all microscale applications. To overcome this difficultly, we propose a passive approach to tune the elastohydrodynamics in a microsystem, towards achieving a pre-determined flow geometry. That is to say, we use the interaction between a soft solid layer, the viscous flow beneath it and the shaped rigid wall above it, to tune the flow domain's shape. Specifically, we study a parallel-wall microchannel whose top wall is a slender soft coating of arbitrary thickness attached to a rigid platform. We derive a nonlinear differential equation for the soft coating's fluid-solid interface, which we use to infer how to achieve specific microchannel shapes during flow. Using this theory, we demonstrate the tuning of four categories of microchannel geometries, which establishes, via a proof-of-concept, the viability of our modeling framework. We also explore slip patterning on the rigid bottom wall of the microchannel, a common technique in microfluidics, as an addition 'handle' for microchannel shape control. However, we show that this effect is much weaker in practice.