How a liquid drop sits or moves depends on the physical and mechanical properties of the underlying substrate. This can be seen in the hysteresis of the contact angle made by a drop on a solid, which is known to originate from surface heterogeneities, and in the slowing of droplet motion on deformable solids. Here, we show how a moving contact line can be used to characterize a molecularly thin polymer layer on a solid. We find that the hysteresis depends on the polymerization index and can be optimized to be vanishingly small (<0.07°). The mechanical properties are quantitatively deduced from the microscopic contact angle, which is proportional to the speed of the contact line and the Rouse relaxation time divided by the layer thickness, in agreement with theory. Our work opens the prospect of measuring the properties of functionalized interfaces in microfluidic and biomedical applications that are otherwise inaccessible.